Apply, do your AIB. If you pass you will attend your initial aircrew medical at CAAMB (Central Admiralty & Air Medical Board) at HMS SUltan. You will do all the standard tests and your "childhood asthma" will be discovered when the Surgeon Commodore looks throuhg your med docs. The Commodore has the final say, but at least he provides an independent case-by-case review of your medical condition. If he does not think that it will be a factor in your flying then you will be awarded an RN flying medical category. I know of at least two front line RN pilots who were in the same position as you. Best advice: give it a go and see what the big man says, nothing to lose really.
Hope that helps.