[shrikant]

If you:

  - are content with a trial-and-error approach
  - have friends/colleagues who can give you useful feedback
  - don't mind making the odd [amusing] mistake

then I recommend going self-taught using a combination of "immersion therapy" (talk to friends/colleagues in as neutral an accent as you can without sounding contrived) and watching a LOT of British TV. This way is slower, but I found this quite fun, and it has the potential for better life stories.

If on the other hand, you:

  - want something more structured and results-oriented
  - don't have enough self-aware friends/colleagues across different British social classes
  - are not in a position to have mistakes brushed off without consequences

then I'd suggest looking up accent coaching. (Although I couldn't possibly recommend any particular ones.)

I briefly knew a Greek girl who'd moved to London for a project management role in a large company here, and she sounded reasonably British middle class (RP) having done daily professional accent coaching for I think 8-10 months.

Sorry I can't be more concrete, but I hope this helps!