I believe the easiest way to achieve this would be to expose yourself to a lot of British media, and also to make it a habit to exclusively use an English accent when speaking English. You may not get it straight away, but you'll find that with time the accent and pronunciation will begin to feel more natural. As for how long, a month would be enough to see definitive results if you work at conditioning your speech everyday. I'm Kenyan, so having overcome my own pronunciation issues, I understand your sentiment of wanting to have an accent that is easily identifiable and understood globally. Plus it just sounds nice :) Ganbatte ne! (^_^)

From your comment, it seems that you see pronunciation as an afterthought, when in fact it should be your main focus. If your pronunciation is in place, then that's most of your accent problem sorted. Input is helpful, but getting feedback is very important. Quite a few of the teachers here help students work on a natural, native English.

Be a chameleon. By this I mean, copy those that you want to sound like. The easiest way is to think of yourself as an actor/actress. You want to play the role of an English person. Imitate the sounds they make. It will feel and sound funny at first but with practice, you will begin to have the accent. PS: It is important to watch how the mouth and tongue move when speaking. Changing the mouth and tongue will produce different sounds. Good luck, Tim