Tim is not intended to take turns, but it's a cool idea
To begin with, I will present why it may be a fine idea to ignore the idea that Tim is not intended to take turns. Next, I'll outline the evidence that Tim is not intended to take turns. Then, I will point out miscellaneous reasons why this intent makes sense.
Time Stop is a 9th level spell, which means your players will be 17th level. Antimagic Field is an 8th level spell, so casting both is a pretty large resource expenditure. At this level, players are very powerful, and while allowing Tim to take his turn may have a big effect, in the big picture, this is not an outlandish ruling and is a pretty cool idea. On TV, it usually seems like some people are immune to Time Stop, such as on the show Charmed.
I can imagine a conversation with an aged wizard who can tell you whether the king is really the king because he experienced a time stop during the critical window during which the king is supposed to have effortlessly removed the sword from the stone. The idea moves me, and I think it is interesting. For me, that is enough to create a new home rule.
Tim is not intended to take turns
Time Stop, a Transmutation spell, says,
So the spell is intended to affect only one person, and to resolve immediately, discussed below. Here is the gloss for Range: Self, which clearly indicates the spell should affect only you.
Other spells, such as the shield spell, affect only you. These spells have a range of self.
Instanenous is defined such that
The spell harms, heals, creates, or alters a creature or an object in a way that can't be dispelled, because its magic exists only for an instant.
In contrast to all the other durations, which each have a beginning time and an end time, and can be interrupted in the duration, including by Dispel Magic and Antimagic Field if the circumstance is appropriate. The word Instantaneous was carefully chosen by the WotC designers, and is specifically designed as a way to adjudicate order of operations for spells.
The way to think about this is that time isn't ticking slowly, it is actually stopped in the universe. The caster exists in a bubble of manufactured time, which is a magical effect. So a person in an antimagic field is affected because the nature of time itself is changed, and Antimagic Field doesn't create a bubble of magical time energy that takes its occupants outside the normal flow of time.
Antimagic Field, an Abjuration spell, also talks about durations:
... While an effect is suppressed, it doesn't function, but the time it spends suppressed counts against its duration. ... Any active spell or other magical effect on a creature or an object in the sphere is suppressed while the creature or object is in it.
That indicates that time works the same inside and outside the Antimagic Field, including instantaneousness.
Finally, Time Stop is a Transmutation spell. Another answer brought this up. If the sun is made to change and burn hotter using magic, if it is turned into a blue star, those in an antimagic field would still feel the heat from this hotter blaze. Whereas if instead, an artificial, magical hotness field was created; the magical field would dissipate when it reached the antimagic field. The distinction is between Transmutation and Conjuration. Time Stop is a Transmutation spell, implying that it actually changes the nature of reality. Thus meaning that those within an antimagic field are not immune to experiencing the side effects of a changed reality.
5e Design Principles - Keep it Simple
At this point I would like to take a lengthy interlude to talk about what makes fifth edition Fifth Edition. In general, 5e is a dramatic simplification from 3.5. Intentionally, it greatly reduces the barrier to entry for new players. It goes to great lengths to ensure that the system is more of an "opt-in" system than an "opt-out" system. The removal of a class or background or replacement of an individual character ability or spell with another does not dramatically affect the overall game.
If you have read all of the spells in all of the official 5e materials, you will notice a pattern. Each spell is written by a formula. In general, a spell does damage, and causes some kind of condition to be applied (such as exhausted), or some kind of game-ified effect (prevents reactions, changes movement speed, allows extra actions). The more damage a spell does, the less conditions are applied. Conditions are always duration limited, and though there are multiple methods, typically a die roll occurs every round until a DC is passed. I think I could reverse engineer a formula, although my intuition says that WotC doesn't adhere to this as an axiom, but uses it as a suggested baseline. My point is, that it is very intentional.
Another pattern you will notice is that the spell descriptions skew towards the formula, and away from imaginative, ambiguous text. There are still exceptions to this rule, but they are far and few between. For example, "flying" doesn't allow you to fly without strings attached. It has been limited just like every other spell - Concentration has been added. This means that any time the target is hit by an attack (possibly multiple times per round), the target has to make a save to end the effect - exactly as the formula indicates. Practically, tactically, this lets the user avoid the 2d battlemap (if any) and position themselves semi-arbitrarily. It also doubles as a wonderful utility spell out of combat, where Concentration is less important. But, in combat, this is an extremely limited spell that cannot stack with other spells.
In general, you will find that powerful spells do not stack under normal circumstances and cannot create a more powerful combined effect. For example, you can't be both flying and invisible through normal means. (Perhaps someone else can cast fly on you, while you maintain invisibility, but that is beyond the scope for my purposes.)
This reflects the same way that temporary hit points don't ever stack from multiple castings. Disadvantage and vulnerabilities apply only once and do not stack. It would be a very lengthy treatise to try to demonstrate this point effectively, so I will not try to prove this unambiguously (it is left as an exercise to the reader). However this intuition is very successful as deciding what the intent is for WotC content.
Instantaneous and Self are both deliberate monikers chosen specifically to stand up to scrutiny in exactly this situation. It would undermine the entire system for, Rules as Written, Time Stop not to resolve independently from Antimagic Field. If another reading was intended, I assume WotC would issue an errata, and add an explicit clause that details its interactions with Antimagic Field, either directly or indirectly. These situational caveats are present in the write ups for other spells.
Other ways in which this intent makes sense
The following section is not meant to convince you of any of the previous discussion. Instead, it is meant to suggest ways in which you can digest this information. Coming at this cold, this rule doesn't make any intuitive sense to me (though, the rules don't have to, and often don't in my opinion). Let us try further to make sense of them.
Antimagic is Lower Level
Antimagic Field is a lower level spell than Time Stop. In the pattern of the various Light and Darkness spells, higher level spell castings of those spells supercede lower level spell castings. This is explicitly noted in each of those light and darkness spells. Thus, if you were to follow this pattern, Time Stop would take precedence, since it is a higher level spell. This could inspire a house rule that a 9th level spell slot expended on Antimagic Field allows the caster to enter stopped time as well.
I think this is a pretty cool idea! You can have your cake and eat it.
Antimagic doesn't negate all magic
Antimagic Field doesn't negate all types of magical effects.
This area is divorced from the magical energy that suffuses the multiverse. ... Spells and other magical effects, except those created by an artifact or a deity, are suppressed in the sphere and can't protrude into it. ...
Admittedly, Time Stop doesn't qualify under this explicit exclusion, which would resolve this question outright. But it does illustrate that sometimes an Antimagic Field doesn't work.
Antimagic is a special spell
... different antimagic field spells don't nullify each other.
DMG, p 41, Crossing the Streams:
... And the famous wizard Elminster of the Forgotten Realms has been said to make occasional appearances in the kitchen of Canadian writer Ed Greenwood --
If you think of D&D as being a mirror universe to our own, where you could walk into your own living room if you had the right spell, what would it mean for you if a mad scientist attempted to stop time here?
Antimagic takes us into a more real world, one more like our own. So if time were stopped here, how would that affect you? ... Whether time can be stopped in our reality or not is a subject of debate, but this more science-fiction take on the question is something I would take into account while reasoning about the effects of Antimagic Field.
In D&D, you traditionally have The Weave. This is the magical fabric of the universe that thread through every location and physicality. Spells take effect by arcane magic users manipulating and moving the threads of reality in one place, such that they cause an effect on connected strands of reality.
(Divine magic works differently, as you channel the power of gods, but you could say that it works differently only because the gods manipulate reality for you, and are absolutely better at it than a wizard; gods exist outside of our time and space (or not!).)
An antimagic field occurs when the threads are frozen in place and can't be manipulated. So, is time transmitted on the threads of the Weave? Does time exist inside of an Antimagic Bubble? If it does, Time must be fundamental enough to exist even where the weave is destroyed or doesn't exist.
If time exists in an Antimagic Field, then the question is - how does it exist in an Antimagic Field. And what is time, and how does time work in your universe. Time may be a single object that suffuses the universe. In this case, the time outside of the Antimagic Field is the same as the time inside the Antimagic Field. And if that is the case, the inside of the Antimagic Field can be affected by changing Time outside the Antimagic Field.
Or, time may be mediated by some other mechanism. As a thought experiment, let us think about another situation - Gravity. I propose two hypothetical spells - Stop Gravity and Suppress Gravity.
Suppress Gravity conjures a force that is opposite to gravity, and in an Antimagic Field, that magical force would be nullified.
Stop Gravity simply disables gravity in the universe. Eg, particles no longer exert gravitational pull anywhere. In one model, gravity is mediated by gravitons. So, perhaps I have changed all the mass in the universe to no longer interact with gravitons.
Now time for some Physics. If Earth doesn't exert a pull, even the few particles that still do located within the area of an antimagic field don't make a difference. Even if Gravity is on inside the Antimagic Field, in physics, if the particles outside don't emit gravitons, inside the Antimagic Field, you can't feel them, because they're not emitted in the first place.
Similarly, if I say time ticks every time the pendulum of a god's clock swings back and forth, and I hold the pendulum back, an antimagic field won't apply. Maybe that's how Time Stop works. Or perhaps we could talk about chronotons.
To me, in order to make the rules make sense in my head, I need to know the mechanism for time. Is Time Stop affecting the outside world, or does it affect the caster? Is the caster suddenly discovering a new, Cantorian, higher order infinity of non-discrete time between any two moments normally inaccessible? Or is the effect radiating outwards, towards the area of Antimagic Field?
I think we have solid justification for the simple interpretation of the rules. However there is also ample room to to allow time to continue for those in an Antimagic Field. Especially since this is just a game, and you should do what inspires you and your group.
It's pretty cool either way, but which one do you and your players expect? I think that the best decision is the one that is the most fun. And this adheres to the spirit of 5e, in that it is **Rules Lite* for a reason. Simple rules as a baseline, and intelligent decision making as an engine.
Explorer's Guide to Wildemount
As @gto mentioned in a comment, EGW introduces Chronurgy Magic and Graviturgy Magic. This late addition to 5e includes Relativity-inspired spells affecting time and gravity, which is very apropos. These spells don't change anything I've already said - these spells must be mediated however they say they are mediated, whereas a different spell would be mediated differently. But it may be instructive to see the terminology chosen - especially for gravity.
Adjust Density - this alters density, which has no basis in physics. It should be called adjust mass. This operates on the target, not on the universe.
Gravity Well - this presumably creates mass, which causes a gravitional well. objects are pulled towards it. The pull would be felt inside an Antimagic Field. If the source were inside of an Antimagic Field, it would not radiate an effect outside the boundaries of the field (if you use physics-based reasoning).
Violent Attraction - increases velocity of blade - this is a magical on the blade. Its title suggests it affects acceleration, not velocity. Also an effect on a target.
Event Horizon - stops people from using movement. This is very thematic, and also operates on targets in a field. If you claim it's a physics based effect, I could see this affecting people inside of an Antimagic Field. However, its type is Graviturgy, not Transmutation, so I could see the ruling going either way.
The Chronurgy spells aren't quite as relevant to the discussion in my mind, as they seem more thematic than mechanistic.
I really appreciate the accepted answer's write up here, which hits the nail on the head. Because the spell is Instantaneous, the turn order does not advance. Additionally, I think this makes sense because Time is stopped, so nothing would be happening while time was stopped. Perhaps it might be thought that all the legendary actions are queued up to be invoked as soon as Annie finishes her turn - but that makes more sense if Time is flowing slowly than if Time is outright stopped.