I would like to pilot or fly in fighter jets however as I have mild asthma I cannot join the Royal Air Force. So is it possible to fly in or pilot these kinds of aircraft as a career without being in a branch of the military?
There's no likely career path, if you have serious enough medical problems they will probably prevent you from getting a job in aviation flying anything commercially. However, that's a separate question, you'd want to talk to an aviation medical examiner to find out what avenues are open to you.
There is a path to flying military style jets as a civilian test pilot, which is open to anyone, however you need a lot of experience and most are retired military.
However, with enough money you can fly just about anything, former military jets do go on the market, often they aren't that expensive to buy, the big costs are maintenance and fuel. If they are to be flown non-commercially the medical requirements are usually lower too. You need to be prepared to spend between \$800-\$6000 per flying hour, depending on the type.