I might have more underground than on the surface. If I injure myself on the surface I am less likely to need to engage in self-rescue and I am probably going to be stuck for a lot less time. If I break my ankle on the surface, it's unlikely a rescue team is going to ask me to hobble off the hillside instead of just sticking me in a stretcher or in the back of a vehicle. There's every chance that hobbling out of the cave is the best solution (both before a rescue team arrives and after).
I probably bring too much crap - it's basically a half Daren drum stuffed full of 'stuff' (including waterproof paper and a pen). Mind you it used to be a full Daren drum which really was too much crap!
On the other hand, not much point bringing stuff if you don't know how to use it - so bring the kit you know how to use.
Paracetamol and ibuprofen are probably an excellent start (for personal use of course); they are good but take a bit of time to kick in so in the event of a rescue it's helpful if you have taken it before the rescue team arrive (but make sure to note the dosage and the time you took them).
Gloves are good for your protection; maybe a few wipes as well for cleaning stuff up (I also have a tiny towel thing).
'Duct tape and a large dressing' (to quote someone else) is also a good idea. You can get small-sized tuff cuts which might be useful. 'Emergency bandages' from SP Services (who sell everything) are basically slightly smaller Israeli bandages.
I also carry cable ties and have used them several times (never for first aid though!).
Steristrips are good but not sure how well they'll stick in a wet cave...
Personally I like having a few plasters just for quality-of-life (and to stop people bleeding over the cave). I have some saline pods but that's mostly because I always wear contact lenses.
I have one SAM splint which could potentially be useful for splinting a broken wrist or whatever (enabling self-rescue) but probably falls in the 'nice to have' not 'really important' category.
Realistically there's not a lot you are going to be able to treat in the cave with any sensible amount of kit. Can't sensibly do much about airway or breathing (beyond ensuring people have their personal medications e.g. epipens, inhalers and knowing where they are). Large dressing covers the circulation (although I guess it's potentially worth bringing some aspirin?). Can't think of anything relevant for disability, which leaves environment which is critical.
Stuff to keep people warm will be very useful. I have a blizzard blanket thing which does take a bit of space and weight but should be much warmer than a survival bag and massively easier to get people into (it's supposed to seal up with tape, which also gives you access when you need it later). I have a bag of skittles in my first aid kit as well (I like skittles). I also have a 2 person lightweight group shelter but that doesn't live in my FA kit (because I may want this without there being an 'emergency').
Never had to use any of the FA kit yet I think (other than maybe a few plasters)... hopefully that continues