As mentioned, there does not appear to be any rules indicating when the fall starts, though it seems pretty clear that the fall is considered instantaneous once it starts.
When does the fall start?
It seems reasonable to assume that it doesn't start in the middle of a jump, or getting across chasms would be impossible as you'd immediately plummet to the bottom the moment your leap carried you over open space. So, this presents a few possibilities that will, of course, be a DM call.
It starts at the end of your move. This makes a good amount of sense as it allows you to complete the jump, potentially clearing pits and chasms. If you don't have a fly speed, then once you complete a jump, you would be forced to end your move. I don't have the rule books, so I can't determine what it says about breaking up a jump like you can other move, but once you start a jump, you are committed - that is, you can't decide to stop mid jump, which could make begin jump-attack-finish jump something that is not a given. Though, see below about using reactions.
Immediately. For reasons mentioned above (clearing pits), this seems unlikely.
At the end of the character's turn. Since move can be broken up, it could be argued that this and number one are functionally the same since you could always choose to move again unless you were completely out of move, so the distinction could be made that move ends when your remaining move is 0, though you might still have remaining actions, making the end of move and the end of turn obviously different. This also overcomes the oddness of being forced to end your move in mid air due to not having a flight speed, falling, then suddenly having move available again once you hit the ground and can again make use of your non-fly movement.
When the character decides. This is probably the easiest to work with. The character doesn't have to wait until the end of his turn to fall, he doesn't have to end his move to fall, he simply falls when it is convenient. If he wants to jump over a pit, he doesn't fall until he has reached the end of his jump and, if his jump is not sufficient to the task, has exhausted all options to try to avoid falling. If there is no help for it, he can decide to fall, then complete both his movement and his turn once he has reached the bottom. The obvious supplement to this rule is that the fall can't be avoided, and so if the creature has not yet chosen to fall when its turn ends, it falls then. This could also allow falling mid jump, so that one could jump to a lower ledge on the far side of a chasm. Optionally, and perhaps advisedly, some actions might need to be declared before the fall begins (or the jump preceding the fall begins) to show preparation and some level of aiming before control was completely ceded to gravity and momentum.
Treating the Fall
Once you decide on when it starts, then you have to decide how to treat the fall itself. The rules say that falling is instant, but really the same can be said of many actions, such as regular move and attacking. Even most spells just have a "duration" of "1 action". We know that a round is roughly 6 seconds, we know that the turns are supposed to be simultaneous, and thus are also roughly 6 seconds, but the length of time that each part of a turn takes is vague. If your character walks from point a to point b, it is effectively treated as instant, and yet other creatures can use their reactions to affect you mid-move. It makes sense that the same would be true of a fall. This is borne out from existing rules such as the Feather Fall spell, or the monk's Slow Fall ability, which allow a reaction to be used to reduce or eliminate damage from a fall (even by a third party in the case of Feather Fall), among other effects (such as changing the speed at which you fall). However, from the rules regarding Opportunity Attacks (PHB p195), you do not provoke OAs when falling, though nothing is said about other reactions, including readied actions, from being prohibited.
So it appears that the falling creature, and other creatures around him, can treat it just as any other type of movement, with the advantage that it does not trigger OAs, and the disadvantage that you have no control over where you end up unless you find some method of stopping the fall (and where you end up very likely comes with some HP loss). But like any other movement, you can take any actions you choose at any point in the movement that you choose, and others can respond to your movement as they would any other form of movement, aside from OAs.
Ruling at my Table
My preference, considering all of the above, is to allow the falling creature to choose when the fall takes place as per option 4. By this interpretation, in your example, the character could then make all of his attacks as he plummeted by the floating wizard (or leapt past him, or landed on him, depending on how he targeted his jump). Again, it may seem unrealistic to take what seems to be a 4-6 second action during what would likely be a half second or so window, but the rules aren't designed to closely model reality, and frequently favor simplicity and flexibility over such realism.
As always, the DM can decide on a case by case basis that even the rules as written (or this interpretation of them) create a preposterous result and disallow it, but this particular scenario does not seem to me to warrant such a ruling, though imposing disadvantage to the PC due to not being able to properly brace himself for the attacks, and also making them hastily, would make perfect sense.