We managed to get from the vets a tablet called kapstar or something similar when "The Alley Cat" arrived riddled with fleas. It was absolutely incredible, within 15 minutes of me having given him it, the fleas were falling off him
Id also pre purchased a nit comb, as while his previous "owner" said she had flea treated him, I saw enough of the house in viewing him to pre prepared, just in case (hindsight is a wonderful thing should never have brought him)
We let him settle in for half an hour on getting home then I cracked out my trusty nit comb, he'd been a bit itchy so I was already twitching and yep sure as hell there were fleas
It was a Friday so I hysterically called the vet asking what they had as I couldn't cope with having fleas present over the weekend, they just needed his weight to give me this miracle tablet, so whipped him up there for a weigh and got home with the wonder tablet
Administer it in the kitchen, and then kept nit combining them into the bowl of boiling water (the fleas not AC!!)
I also got one of the flea sprays they sell and blitzed the lounge as he'd been exploring for a bit on getting home before I did the nit combing. Hot washed the blanket that was in the carrier he'd been in.
Until you can get an appointment tomorrow, I'd be nit combing and putting anything you get out in a bowl of boiling hot water. I'd also consider shutting kitten in the kitchen overnight so the fleas don't have any soft surfaces they can get on, and getting a proper spray when you go tomorrow (he'll have to be shut upstairs while it airs out after spraying if you have more than 1 floor). Nit comb was a trick I remember from my parents getting a Moggy and her also apparently being flea treated but my mum having to sit on the kitchen floor with her bowl of boiling water and nit combing her until she could get an appointment for proper flea staff
I have a picture of the bowl from when we did AC, it's absolutely vile
Definitely ask for Kapster (I think it's called or very similar) it's absolutely blooming brilliant!