Admittedly this was all some years ago, but...
I was binned from an RAF Flying Scholarship cos my Dad insisted I admitted to having had asthma as a child, though gone since puberty.
Three or four years later I joined the RN as aircrew and they didn't bat an eyelid over it.
Three years later (1985) I tried to get a Class1 at CAA house, but the Senior CAA examiner refused it....due to "childhood asthma" and the results of a peak-flow test! No amount of reasoning that I'd been through all that Naval training, wings etc and passed fit to fly FJ at 40,000', nothing would change this fool of an ex-Crab's mind (for that's what he was).
I got a friendly avmed doctor on side and all I had to do was to learn the correct technique on the peak flow meter (technique helps on these things) and after a violent 10 minute yelling-and-shouting argument with Chief Doc Crabtwat and actually putting his career on the line the junior CAA doc I'd seen for the retest persuaded the blinkered old buffoon there was no reason to deny me a medical. I wish I knew that Doctor's name so I could thank him.
Clearly the RAF still has a bug up it's arse over Asthma, but good peak flow is usually considered a clear indication of no Asthma history.
But you never know with stuffy organizations - so try the Navy too. It's a damn sight more fun anyway, or go for a civil career, but avoid the ex-RAF doctors on the medical board!