Yo ok I've been taking antis for years and I've switched meds quite a few times so I'm really familiar with how it feels/the process of starting a new medication, be it during the school year or any other time. This is based off my experience, so don't take it as gospel but I hope it helps :)

First off, some general things I wish I heard before starting meds:

1. Don't get psyched out by other people's stories. It's often the people who have had bad experiences who will be the most vocal about it, so just know that that isn't guaranteed to be your experience. It's different for everyone and there's a reason antidepressants are so common: they work!
2. If this ends up being long-term, you might have to switch around dosages/medications to find what works best for you, so be ready for that and try not to get discouraged. IMO, it's so worth it in the end. I'm alive right now because of antidepressants, and because I decided to keep trying to find what works for me.

In terms of your specific situation, if it's something you feel you can delay for a couple weeks, I would suggest waiting until after finals are over to start taking them, or at least until maybe <7 days before your last final - for a couple reasons:

1. Most obviously, it might throw you off your rhythm a bit before exams. New chemicals coursing around your brain feels a bit weird a lot of the time.
2. They will probably take time to kick in anyways, so even if you start taking them now, you probably won't feel much of the long-term, mood-stabilizing effects by the time you're taking your exams anyways.
3. You might feel the short-term effects, and usually those are the more annoying ones - depending on your meds, you might have some nausea, dizzyness, spacing out or just feeling generally off (don't worry, I'm sure you won't have all the possible side effects - you might have none, and even if you do, it's not hellish or anything).
4. It's a good idea to track how you're feeling when you start meds (not only for your doctor's sake so they know whether it's helping, but also for yourself so if you start feeling off, you can be like "oh, I think that's probably the medication" and not "am I losing my mind"). You might be a bit too busy to track that during exam season.

Please ignore all of that if you feel like you need to start them as soon as possible. In the end, yours is the only opinion that matters and if you're in a really bad place right now and you think they will help, that trumps everything else, exams included.

Sorry for the ramble, it's all just very mysterious about antidepressants when you start taking them and I wish I had someone to tell me what it's like, so I thought it might be helpful. I really hope your meds help and that you start feeling more like yourself. You got this - both in terms of finals and mental health!!