Pathfinder has no rules for crippling injuries
That is, there are no rules for what happens when you are crippled, because there are no rules for being crippled in the first place. Crippling injuries are simply not a part of the ruleset. Anything crippling a character, and any effects of this crippling, are purely a matter of houserule because the Pathfinder system does not support either concept. (It does, oddly enough, have a single mention of healing crippling, in the regenerate spell, which is pure legacy since the regenerate spell is quite old, older even than the D&D 3.5e that Pathfinder is based on.)
There are optional rules for called shots that can result in “debilitating blows,” which can include crippling injuries. As an optional variant rule, these are basically some houserules that Paizo decided to write up and publish. They are, in my opinion, very poorly designed (based on a casual skim; I have not used them but I don’t feel that I need to based on what I’m reading). At any rate, extremely limited information is given on losing a hand, so even if you were using them, you would still need a lot of GM ruling to decide how losing a hand actually works.
Anyway, houserules and optional rules variants are generally seen as the GM’s responsibility. If he was changing the game to include ways of becoming crippled, it was also his responsibility to determine what those were. Furthermore, since this is a massive departure from the rules as they are presented officially, I would consider it his responsibility to explain this change, how it can happen and what effects there are when it does. Since I play Pathfinder in part because I am not interested in this kind of thing, I would decide at that point that the game wasn’t for me and graciously leave. If it was, instead, sprung on me, I might be less gracious about leaving.
Any reasonable houserule for a lost hand make the character unplayable
So with all that in mind, my first thought is that a crippled character is effectively out of the game. Repairing a crippling injury apparently requires regenerate, a 7th-level spell—restoring the hand is more difficult in Pathfinder than raising the dead. Absurd, I know, but then Pathfinder wasn’t designed with this kind of thing in mind. Unless regenerate is available, the crippled character is basically forced into retirement and is now an NPC. The player rolls up a new character. In fact, the only time I would consider having a player character be crippled in one of my games is when a player wants to retire a character, and move on to a new one.
That’s the easy answer, but it doesn’t always work narratively. Maybe you aren’t in a safe place to retire; after all, you just had your hand chopped off, that’s not exactly a hallmark of a secure location. So what happens? We need houserules. The ramifications are largely pretty obvious here, but the problem is that they make the character largely unplayable. At the very least, the drastic loss of capability makes the character far, far weaker than he-or-she should be at their level—which will make it nigh-impossible to continue adventuring with the same party. He-or-she is now a liability and will need to be carefully protected by allies to ensure survival. Specifically...
You only have one hand—you can’t use two hands on a two-handed weapon, you can’t use both a weapon and a shield, you cannot use two weapons at once. Note that those three encompass literally every viable physical combat strategy in the game.
Furthermore, you can only cast spells with somatic gestures if you are not using any weapon or shield, which is a pretty serious repercussion. You likewise cannot do so if you want to use any other kind of gear or magical equipment that needs to be held in the hand. Those are very big deals, though spellcasters enjoy enough advantages in Pathfinder that this is probably workable, at least short-term.
You also lose the hands slot too since almost-all hands-slot items require a matched pair, which you cannot wear any longer. You might also lose a ring slot—the rules only ever say that you are limited to 2, without really specifying that they must be each be on a finger and not both on the same hand, but that is certainly at least the normal expectation, and often this is suggested as part of the logic for why rings are limited to 2 (specifically, it is common to suggest that magical rings interfere with one another so you must separate them by putting one on each hand—and actually putting it on a finger is also mandatory).
At this point, we have a character who either literally cannot do what he-or-she is built to do (if they used any kind of physical combat style), or is “merely” at an enormous disadvantage (if they relied on spells) since even if you didn’t “need” both hands, the loss of basic functionality is a huge blow to your capability. This is why the character basically needs to retire, barring regenerate.
And we haven’t even gotten to the more nebulous ramifications, particularly skills. Adjudicating skills is probably going to have to be a case-by-case fiat by the GM.
Note that the above, abbreviated though it is, nonetheless represents a more thorough treatment of the ramifications of losing a hand than Paizo’s own optional rules, which just say that “the hand is severed or otherwise mangled such that only regeneration or similar effects can repair it. Regardless of the result of the saving throw, anything held in the wounded hand is automatically dropped, even items held in two or more hands.” However, it’s worth noting that this situation only happens under the called shots variant if a called shot results in a debilitating blow that deals at least half the target’s total hp (minimum 50), and the target fails a saving throw. Note that another optional rule, the massive damage rule, would have that save actually be against dying outright (which, in most cases, would be preferable as it’s more-easily healed ⌐.⌐).
The situation can be somewhat improved
The above are the reasonable, expected results of losing a hand. They make the most sense, based on what characters use their hands to do.
They also make the character nearly unplayable, and that’s no good.
If we decided, then, to design our houserules around minimizing the disastrous nature of this storyline, rather than necessarily what makes the most sense, we can do somewhat better.
First, it makes absolutely no sense that restoring a hand is massively harder than restoring life itself. Adding the ability to restore a hand to the effects of restoration, or maybe even lesser restoration, goes a long way towards improving things.
Second, definitely should have some way to rig a glove (and ring, if necessary) to your arm-stump. Losing those magic item slots is just unnecessary. That’s easily imagined, anyway.
Third, I would consider also allowing some way to rig a shield there. Yeah, a wizard might not use a shield—but mithral bucklers have no arcane spell failure and no armor check penalty, so a wizard can use them with no problems, and should do so. The AC bonus is minor, but the ability to have another platform for magical properties is a big deal. Even if the wizard doesn’t use a shield yet, they will want to eventually.
Fourth, while this doesn’t apply to the wizard so much, for other characters it becomes necessary to improve the viability of an einhander (one-handed) fighting style. The duelist prestige class is atrocious, unfortunately, and at any rate even if it weren’t, the person who loses a hand probably doesn’t have levels in it. Some kind of retraining option is a good solution to that problem (though it necessitates a substantial pause in the quest, which may not make narrative sense—magical retraining may be a necessary answer there), but only if there is something decent to retrain to. The duelist class is not that. We have a question about better duelist classes, which is for 3.5 but might be a decent start to figuring out how to handle things (though I would probably adapt it, probably making its features available to martial classes as a kind of archetype or something).
That at least puts you in a far better position. It’s still a really big problem—there just isn’t enough material for handling crippled characters well and a Stack Exchange answer is not the place to develop everything you should—but it should at least be playable. And ultimately, even if you give the optional called shot rules far more authority than they deserve, this can still be compatible with them—those rules give extremely little detail on the full ramifications of losing a hand and the above could fit those just as well.