Will double post with something approaching real advice.
The first hurdle you will have is that you will need to clear customs before they let you run amok in London. Usually, if you are polite, explain what you are trying to do, show you have a plane ticket out, etc., you won't have many hassles. What will be a potential hassle is the time needed to actually get to a customs agent. This is somewhat random and depends on how many other foreign jets landed at approximately the same time as you. I (this was at Gatwick) once went through on a Wednesday afternoon and it took me about 30 minutes. Another time, on a Monday morning, it was nearly four hours. Your mileage may vary but this might be the first go/no-go decider.
Next decision, once you make it through customs, is to exchange any money. Britain still uses pounds as currency rather than Euros like the rest of the EU. If you are adventurous, you can probably get by using only an international credit card like Visa or Mastercard. (Make sure you give your credit card company a heads up as they can deny charges if they think the charges are 'suspicious', that is suddenly appearing in London when you were just in Indiana. Usually a call to their customer support is sufficient.) If you do decide to exchange money, you can sell back what you don't use. Just be aware that you will probably lose juice on the exchange both ways.
Now, through customs and money problems decided, next decision is if you want to tote all of your carry-on crap with you. If you want to stash some things for the day, look for a 'Left Luggage' place for storage. Remember to allow time to retrieve your stuff on the return leg.
To get from Heathrow into the London metropolitan area you will need to use the British Rail system. There are express trains from Heathrow to London Paddington and I would recommend them if you are on a 'time is of the essence' mission. The local trains will cost less but take much longer to get you there and back. The train terminal at Heathrow should be easy to locate by reading the signs, but may be a semi-hefty walk.
Once you are at Paddington you have access to most everywhere through either the underground trains or buses topside. You will have to check, but I would guess that city tours start from either Paddington, Victoria, or Waterloo stations. Possibly all three, but you will need to do the research on that. If you decide to go to a location rather than just tour about you need to figure out if you want to go by bus or underground. The underground, in general, is all sorts of convenient. The problem is that your view of London is a series of tunnels until you reach your destination. Travelling by bus gives you more chance to see the city, but London street traffic can be brutal and delays are quite possible.
Remember to allow sufficient time to get back to Paddington, get from Paddington to Heathrow, collect your luggage, get through security, get through outbound customs and get to your gate.
Hope this is somewhat lucid and helpful.
-----signature-----
-Bahgoon
