+1 Know many who have done that on way to USA from India. (Note that it is still dependent on what particular immigrant office considers "risk". So be extra polite and well behaved even if LHR immigration line is a zoo )
If not familiar with London, just take tube to Piccadilly and take one of those hop on hop off bus (get off only at few select stops like buckingham palace, bigben, tower bridge)
Warning: Tube back to airport can break down or get severely delayed. Plan to reach Airport 3.5 to 4 hours before flight. Security lines can get crazy too.
Given the time it will take to go through security, reach central london and back with 3-4 hours buffer before flight, you may have barely time to do full round of HOHO bus without even getting down in which case, I wud just buy the day ticket on tube (remember to collect it after putting through machine - with day ticket you dont need Oyster card) and take airport line to Hammersmith, Change to central line (which is on tracks across on same platform) and get off at Westminster. After strolling around to see Abbey, BigBen, crossing bridge for better views of Parliament, London Eye, either walk or take bus to Trafalgar Square and from there to Piccadilly and then head back. If you have more time then walk the Mall to Buckingham palace and then take tube back from nearby Green Park station.