Hi and welcome.
Well, I'm based in NI so I know a few different folks who have trained in London. Certainly the big, sexy gyms (like Roger Gracie's Academy and Carlson Gracie Team London) are pricey.
That said, I remember reading a quote from the guy who co-runs the Carlson Gracie Team in London and he said the reason they charge what they charge is they want to build a successful team, and to do that you have to have people who are willing to make a commitment with their training, not just somone who shows up once a week.
I'd say if you're looking to try martial arts for not big money in London, check out the Judo clubs, except the Budokwai, which can be a bit pricey (although the instruction is world class). Alternatively, look at boxing clubs. Both boxing and judo are Olympic sports and get subsidised by public funding in the UK. The main knock on effect of this to grass roots clubs is that you tend to get charged less for class.
I wouldn't discout other sports though, especially rugby. People tend to think physical skills learned in MA classes are the only thing that are worth knowing in a fight. Not true. A young actor who was in a Harry Potter movie was stabbed in London a few years ago. One of his friends, a rugby player, rugby tackled the assailant to the ground so hard he knock the knife out of his hand. There are a lot of useful skills like mental toughness and fitness you get from rugby too. So don't count that out! Matt Stevens, the Saracens player, got his BJJ blue belt in 3 months because he killed everyone in BJJ class because of his rugby background. Normally it takes 2-4 years to get a blue belt in BJJ.
Just some food for thought.
"Let your food be your medicine, and your medicine be your food"