As said, what exactly do you wish to work on as an engineer?
If your desired field is aircraft, then like pilots, it makes more sense to join a service where there are more places, and that would be the RAF. If you want to be a pilot, what would you like to fly? You won't get a choice in the discipline of aircraft you train on after successfully completing EFT, but it makes a lot more sense to join the Army if you want to fly helicopters, or the RAF if you want to fly multi-engine aircraft. You can fly fast-jets in both the RAF and RN, but there are a lot more current fast-jet pilots in the RAF, and there are currently no frontline fast-jet pilots in the RN. Helo's are flown by all three services, but as I said, you won't get a choice unless you join the one service that only flies helicopters, operationally.
If you suffer with asthma or have done before and it's on your record, you can forget being a pilot in any of the three services if you are unsuccessful in getting asthma written down as a misdiagnosis on your medical record. There are lots of things to consider, and I wish you luck as I too would love to be a pilot!