crimson_twilight: Let's try an oldie. Hugo 3: Jungle of Doom?
Penelope gets bitten by a spider in the opening cut scene and you're searching for a cure.
(The trilogy was also a form of shareware, so it could easily have been installed on school computers)
https://www.gog.com/game/the_hugo_trilogy I had a look at this one after someone suggested it on Reddit, but the colour scheme definitely wasn't 3-bit(? if that's what this is, you know what I mean anyway, it wasn't just primary colours). Thank you though!
skeletonbow: Sounds like id Software's DOOM, the evil robot spider motherbrain.
Definitely wasn't DOOM hahah, I think parents would have called up in anger if they'd installed DOOM on the school computers, but thank you!
Caecity: Not this - but this is almost exactly the colour scheme I remember! I believe the game I'm looking for was closer to a puzzle game than a platformer, I definitely remember seeing some kind of 'map' screen (to show places you could go to). Thank you for the suggestion!
Thanks everyone for your suggestions so far - I feel like I must be getting closer to finding it after all these years, and again, so sorry for my delayed reply :(
BreOl72: Have you also tried asking over at "The Escapist"?
They often manage to name old games, after vague descriptions alone.
:) https://v1.escapistmagazine.com/forums/read/9.166417-Old-Games-you-remember-playing-but-cant-remember-the-name-of?page=933 Yeah, I came across that thread actually! For some reason my email wasn't receiving the account verification thing, so I tried to contact their support. Still no reply though, so I think I'll try another email provider with a new account and see if that works