If you do think you can get into central London, the fastest way is by Heathrow Express train which takes you to Paddington in about 30 minutes (but remember buying tickets time, waiting for the train time, etc).
You may prefer to consider a private tour in a black cab which could collect you from the airport and start your London tour immediately. I went out for a tour with Graham Greenglass of London Cab Tours and can recommend him.
From there you can get the London Underground system – Bakerloo Line (the brown line) – to Charing Cross. This is the station for Trafalgar Square. This has some great photo opportunities. From here you can walk down The Mall (one of the main roads from Trafalgar Square) where Buckingham Palace is at the end. The Changing of the Guard ceremony is at 11.30am each day but even if you miss this it's still fun to see the guards and the Palace.
Head for Horse Guards Parade at the other end of St. James's Park and walk through the archway to see the mounted Household Cavalry. These are part of the Queen's protection team and again, make great photos. Walk through the main road, which is Whitehall, turn right and you'll see 10 Downing Street, where our Prime Minister lives. You can’t get close but you can just see the door from Whitehall.
Walk to the end of Whitehall and you'll come to Parliament Square. Here you can see the Houses of Parliament and Big Ben, plus Westminster Abbey. Go onto Westminster Bridge and you'll see the River Thames. Look left and there's the London Eye – an enormous observation wheel and a major landmark on the London skyline.
Now, to see this much you'll need at least a couple of hours but you will have taken in some of the really important London sights.
See a Google Map of this itinerary.
I wouldn't recommend going to the Tower of London as well as it's a bit further down the river (towards the City of London – the old part) and the entrance fee is too steep to not spend a whole day there.
If you finish your whirlwind tour at Parliament Square you can go to Westminster tube station and get the Circle Line (yellow line) back to Paddington to get the Heathrow Express back to Heathrow Airport.
I think this would make a great introduction to London and I hope you’ll able to give it a go. I would say always allow a little more extra time for getting back to the airport than you think as tube train delays do sometimes occur.
And the good news is all the things I've suggested you do in your few hours in London are free.