Upvote:7
Since the UK doesn't operate exit checks you can just buy a ticket and fly home. Airlines give passport information about their passengers to the authorities in advance, but the UK's main interest at this point will be to see you gone, so they have no interest in trying to stop you from leaving. Do not expect to be let back in anytime soon (no matter how you leave).
You will not get any particular benefits out of trying to contact UK authorities directly before you leave.
There are many airlines that operate direct flights from London to the US, so you should not have any particular problems finding one. However, if you find a bargain deal with a layover in Amsterdam or elsewhere in continental Europe, that shouldn't be a problem for you either.
American citizens do not need any visa to transit through the Netherlands or other Schengen countries. If you're only changing planes, you won't even come into contact with immigration authorities at all. The airline will want to check that you're allowed to transit without an airport transit visa, but a US passport will more than sufficient to satisfy that need, no matter what your travel history is.
I'm not sure about Ireland though -- the Common Travel Area works in non-intuitive ways sometimes, and I think you'd actually need to enter Ireland in order to change planes there, and need to pass through an Irish immigration check. If so, your history as an overstayer may create problems for you at that point. Personally I'd steer away from that risk, given the plethora of other possibilities.