And if you want to be slick, just use the e-gates with your USA passport. No one even needs to know. Not the "by the book" advice we typically give but as long as you aren't travelling with children under 12, you can use the e-gates.
The Home Office will know as they used the e-gates with a US passport. The exit checks would also show that visitor didn't leave before 6 months. The exit checks were brought back in to find who wasn't leaving the UK, but it is also valuable data for the Home Office.
The UK records entry and exit into and out of the UK.
The airlines have to give the UK the names of the passengers on flights: the border officers at ports inform the HO too, as do the UK border officers in Belgium and France. It's how the HO/Embassy staff are able to meet certain passengers as they arrive at the border/try to board a plane to the UK or enter the UK at a port and how the HO prevent certain people from leaving. It's also how the new law was able to come in last year, to not to have to give notice of removal of British citizenship, so that when the exit list flags a name, they can just remove their citizenship and take their Brtisih passport when they try to enter the UK.
As long as the airline will let you board and you can get to the UK border, the UK border guards cannot stop a British citizen from entering the UK. They might not be happy and there might be a delay.