Please find attached the rules from the RN website about medical conditions and entry into the Naval services (Both full time and reserves)
The Armed Forces require anyone who joins to be medically fit to serve worldwide. New entrants to the Armed Forces undergo intensive training which is physically demanding and mentally taxing. New entrants must be of a robust constitution and free from disease or pre-existing injury to meet this challenge. If you do decide to apply and your application is accepted, you will be required to pass a full medical examination. Your application will be rejected if you fail to meet the minimum acceptable medical standard for entry. Before a Medical Officer examines you, you are asked to declare certain information about your medical history to identify obvious medical conditions for which rejection is automatic. Please note that your application will be rejected if:
You suffer from Epilepsy
You suffer from Diabetes
You suffer or have suffered from Asthma in the past four years
You suffer from Colitis, Crohns Disease, or any other long standing bowel disorder or have had your spleen removed
You have ever had a detached retina or corneal transplant
You suffer from Rheumatoid Arthritis, or any other bone or joint disorders
The above list is not exhaustive and you may find that your application is rejected due to other medical conditions, not listed, but which are subject to the discretion of the examining doctor or Service Medical Authorities.
Hope this helps in understanding the Forces medical requirements
Sorry it's not good news, but best of luck in your future
As Supermario says, asthma is a bar to entry in general terms. There are a couple of exceptions, the most important of which is the use of medication. If you haven't needed any for the past 4 years (ie inhalers, steroids, longer-acting agents) then you may be considered, but it is subject to medical assessment.