Format is: width, rolling diameter, wheel diameter.
Rolling diameter for 195/50/15 would be (25.4*15)+(195*50*2/100) for example. That was (inch in mm*wheel diameter)+(width*sidewall*2/100%). Rolling diameter is pretty vague tbh (e.g. the same size between two brands can easily be 10mm different) so just get something close.
Most are in mm (Dunlop, Pirelli, Avon)
E.g. 280 660 R18
But some (Michelin) are in cm:
E.g. 28/66-18 (size doesn't exist btw, just showing what the above example would be in Michelin)
Clio cup UK cars ran 19/57-15 Michelin. S9 (hard) compound.
Compound codes are hilarious. Dunlop & Avon are a seemingly random string of letters & numbers that you need to look up on a table to know what is what. Michelin have a 3 digit code where the middle number, 1 to 9, denotes soft to hard so you can work it out yourself. Pirelli are my favourite as it's really simple: the first letter is D or W (dry/wet) and the second is S, M or H (soft/med/hard).
As for where...
New tyres - just search Avon Motorsport etc and look up the distributors details. HP Tyres are popular for dunlops.
Used tyres - find a one make race series that use a suitable tyre size & then agree cash on collection with a local team. Lots just put them on eBay these days tbh so that's worth a search. Most UK series use Dunlop atm so they're probably your best bet. Bound to be people running 15" in Britcar for example. Someone on here might know someone who knows someone who knows someone. I can't help with 15s.
What do you want them for? Just to go quicker on track days? Any used slick will be massively better than any R888/Dirreza etc, but ones which have been very well looked after will be worth another 2 seconds a lap. If it's just track days u wouldn't get caught up in worrying about how many heat cycles they've done etc, just make sure they've got some meat left on them.
P.s be prepared for generic comment about needing mental spring rates, aggressive pads & spherical bearings all round to get the most. Oh and probably something about less grippy tyres teaching you to be a better driver