I lived in London for a one year. It was the time when I got interested in the british accent too.
I remember myself traveling in the underground, listening to a podcast. Hours, many many hours. I think before you will sound like a brit do, you must listen intensively authentic, spoken British English.
I suggest you to check out Luke's English Podcast - It's awesome for that purpose.
Another good way to improve your accent is as I call it - shadowing. You play a bit of speech or a movie, you pose and repeat after a native speaker. Try to pronounce sounds the same as this person does. If you're already quite fluent, you can also play that bit without pausing, in a little bit slower speed e.g. x9 and speak with that guy.
Speak to yourself and exaggerate while doing it. Find out what features of the british accent are and learn them. You can check out ETJ English on YouTube. You'll find a ton of useful tips about british accent there. I think that quite important thing is to learn how connected speech works. You'll sound more nature.
And of course practice whenever you can. Don't by shy and make use of any interaction with brits to practice your accent. But you also need someone who can correct you. Recording of you speech and listening it back will often point out sounds and mistakes you need to work on.
And even more importantly - be passionate about it. Find those sounds, the melody of this language pleasant for your ears. You'll be much more tuned to details and make greater progress. I promise !
I hope it will help. I go back to the memories from London when I used to live there and I feel a bit jealous of you ;)
Really cool city, although a bit crowded ;p