Do you go thru Border Control in London if you’re only stopping there for a connecting flight

Doesn't matter if you are a US citizen or not. You will probably go through security again but not immigration if you are just connecting. Assuming your flights are both on the same ticket, that must be considered a legal connection, where you can connect to the second flight without crossing into the UK.

You should check to see if you need a transit visa to pass through the U.K. I only mention this because you do not say what country issued your passport.

I can only speak from my experience. I flew to Vienna via Heathrow on two separate airlines. I did NOT go through immigration; I went through security.

You can connect “airside” and not go through immigration/passport control. You will go through security to reach the gate for your onward flight.

Lola makes an important point. To clarify what i wrote above, I stayed airside and did not need to go landside as I arrived & departed from Heathrow's Terminal 3 which is why I only had to go through security.

You can stay airside even if transferring between terminals. Just follow the purple “ flight connections” signs for the appropriate terminal.

British Airways uses both T5 and T3 so you may or may not have to change terminals. But the security screen will take place regardless.

To see all the steps involved as well as the estimated time, go to the Heathrow connection tool and put in all your flight details: https://www.heathrow.com/flight-connections