Flight from U.S. through London to Edinburgh

You go through passport control and security at LHR. You make your way to your connecting flight.

Details here https://www.heathrow.com/flight-connections

Your luggage is checked through from the USA to Edinburgh where you retrieve it and pass through customs. Customs is a non-event unless you have goods to declare.

most excellent information. thanks.

You do have to reach security in Heathrow Terminal 5 minimum of 35 minutes before the scheduled take off time. If your incoming flight is delayed you will be put on the next available one.

You actually have a choice of different sets of immigration facilities - either going completely landside with normal arrivals, or doing it airside through the 'flight connections centre' or FCC. The latter is shorter in terms of distance walked but whether it is shorter in processing time is not clear cut.

Be aware that through-checked bags in Edinburgh may come out separately from those who have taken this just as a domestic flight.

Definitely go the flight connections route. When you exit your plane upon arrival in London, you will see Purple "Flight Connections" signs with arrows. Follow these well-marked signs. Eventually you will come to a point where the Flight connections split into two parts. The first line is for flight connections to the United Kingdom and to Ireland. (Take this one!) The second line is for international flight connections. You will go through passport control, have a biometric reading done that associates your picture with your boarding pass (takes about 15 seconds) and will then go through security screening again.

I've done this type of flight connection twice. The first time, we were a little pressed for time. We simply told the attendant and we were wisked to the front of the line at passport control to make sure we made our connection. The second time we we had plenty of time and just stayed in the line as it moved along. Passport Control took 10 minutes, Biometrics less than 1 minute, and security about 10 minutes. We were then ready to go to the next gate. The number of people trying to go through at once will certainly impact the time spent. Our experiences have been positive.

One hint is to make sure that your bags are completely free of water bottles, etc prior to arrival at security. People will wait until they go through security at their home airport and then buy a bottle of water and put it in the bag without thinking much of it. However, if it is still there when you go through security at Heathrow, you will be spending additional time as your bag will be sent to an alternate line where you much wait for a security person to look through it with you. Keep your liquids in their 311 bag and ready to pull out.

One feature of Heathrow is that they do not give out gate assignments far in advance. When you look at the departure board for your next flight it may say something like "gate information will be given out at 1300". Keep checking periodically because more information may be coming prior to the actual gate assignment. It may start to tell you which terminal (A,B,C) that your flight is leaving from before it actually tells you the gate. As soon as you know which terminal, start moving that direction.

Your bags will only be checked through to Edinburgh if both flights are on the same booking reference number.
Yes 'flight connections' - the purple signs, is the way to go - do NOT exit through customs in London - this will add a lot of time and hassle. My recollection is that you will clear immigration as part of the connections process. Your 3 hours should be plenty of time to clear both immigration and security, although queues can be long!
Yes - ensure liquids conform to the 'one small zip lock bag' policy. You will need to remove laptops and other large electronic devices from your bags, and remove jackets. There is no need to remove shoes at security - unlike in the US...
Be aware that BA internal flights normally depart from the A gates in Terminal 5 at LHR, so that is the main part of the terminal. Do not go to B or C gates unless the departure boards show your flight as leaving from there as it's a long way and will take time to get back to the A gates area.

Skyegirl makes a valid point, and while it might not apply to you, last summer we had separate bookings on BA from Denver to London Heathrow, and from Heathrow to Glasgow. We asked when checking in in Denver if they could somehow get our bags checked through all the way to Glasgow, and the check-in agent did a lot of typing on her keyboard, and made it happen. Sounds like we lucked out, and another time with a different person, we might have had to claim our bags in London and re-check them for our flight to Scotland. Our connection time was much tighter than 3 hours and that might have been a factor.

Also, the immigration line for getting our passports checked once we landed at Heathrow was long and slow, and with our short connection time, a UK Border Force agent was nice enough to move us up to the head of the line, so we could make our connecting flight. Hope your transfer is smooth and not as slow as your past experience.