I have a new cat (here he is! and he is very keen on licking my face. He particularly likes to lick my top lip and often tries to stick his nose up nostril. I understand why he’s doing it - he’s marking me with his scent. He also likes to curl up on my chest with his face very close to mine.
I don’t mind it because he is small and lovely, but I do worry that I am breathing all over him and he is breathing all over me. Should I be worried about germs?
Awwww submit him to www.ratemykitten.com
The technical term for “diseases people and animals can catch from each other” is “zoonotic”.
Diseases you can catch from your cat.
Diseases you can catch from carnivores in general, including cats.
Myself, I wouldn’t, but then, I’m not a “cat person”. 
I mean, I wouldn’t worry about germs, but then, I don’t have a cat.
Now that you’ve heard all about what you could possibly catch from a cat, I’d like to add some anecdotal evidence. I’ve owned, slept in the same bed as, cuddled, cleaned up after, and been licked by cats since I was five years old, and I’ve never caught anything from one of them. My children have each done the same since they were around two (whenever they outgrew the crib), and they’ve never caught anything. My husband did the same from the age of six, and he’s never caught anything. I could go on, but I won’t.
If your cat scratches or bites you, keep a close eye on the wound, because they can indeed get infected quickly. Aside from that, the only thing on this list that I worry about catching from my cats is plague, and I don’t think you’re in much danger in your part of the world. When I was pregnant, I made others change the litterbox because I could, but realistically, if my cats had ever had toxoplasmosis, I’d have been exposed long before I got pregnant and I would be in no danger.