U.S. citizen flying with visa to visit India. Possible 14 hr. layover at Heathrow. Would like to take train into London to visit family during layover. What is the procedure for doing so? Lived in England for a couple of years, and know my way around, as well as the time it would generally take to do the things I would like to do while there. However, do not wish to have difficulty with airport authorities for leaving and returning. What are the rules for doing so?
As a US citizen you don't need a visa to visit the UK. As far as I know, you can simply go through immigration and customs like any other visitor; all you need is your US passport. You might be asked how long you're planning to stay, and where you're going next, and you'll just say "Leaving for India later today" or whatever. I'm not aware of any special rules that would govern this, and the fact that your onward flight is on the same ticket shouldn't make any difference. It's an ordinary visa-free visit to the UK that just happens to be very short.
Note that your checked luggage will probably be checked through to India by default, so you won't have access to it in London. If you want to have it in London, you may be able to ask the airline to "short check" it.
What are the rules for doing so?
Briefly, they say that a US citizen does not require an entry clearance, just a passport. In order to determine if you are a genuine visitor, you will have a landing interview with an Immigration Officer. As to what the IO will ask you, we already have an answer for that. I suggest you read over it. If things go very, very badly, you will be bounced. You can read this article prepared for an American who had that happen.
When your landing interview is successful, you will get a 'leave to enter' stamp in your passport. Someone has already asked what this means in a legal sense and we have an answer here.
However, do not wish to have difficulty with airport authorities for leaving and returning
Reading the articles mentioned above will give you detailed information about what to expect, and you will (or should) have a smooth transit through UK immigration controls.