From memory, current asthma is a complete no-no for pilots in the RAF. If you were asthmatic as a child but have grown out of it then I think you need to be symptom free for 5 years with no treatment to be considered. The CAA rules will be more relaxed.
Generally, if it's well controlled and the Peak Flow is OK then you may well be lucky.
Certain asthmatics are worsened by exercise, just the same as some are made worse by cold or certain allergens (animal fur, house dust mites, etc) so that is why the CAA make you run round the block!