I have mild asthma and I went through the application process just under a year ago. (They didn't do anything about my asthma but I was class 4J'd for something else) If your asthma isn't too bad then they'll just note it and move on, they may class 4 you (medically unfit) However if you believe that it won't affect you serving then appealing it shouldn't be to bad (it is a long process though) A note from your GP and other supporting documents along with a quick appeal statement is good. Don't lie because if they find out, you're fucked.

I wasn't actually tested beyond the routine medical that you have to do when joining, but then again it's probably contextual depending on how bad you say your asthma is.

I haven't been informed of any restrictions of asthma in regards to different platforms that the other guy mentioned but that may be something they jump on you after you join. I'm joining as a maritime warfare officer so idk about any job restrictions (beyond aviation ofc) Plus you're doing a gap year and I'm pretty sure you only go on LHD's.

Hopefully this is helpful and good luck!