American overstayer in UK going home. Questions about transiting through Schengen Area on the way back

if I make a booking that requires me to change terminal in the Schengen Area airport, would that mean I would need to pass through immigration

This depends a lot on the airport:

• Some airports only have a single terminal, but even then, do not have airside transfers (I don't have an example in the Schengen area, but IIRC Luton in the UK is in this situation, so it wouldn't be surprising if other EU airports had similar arrangements, mostly smaller airports with lots of LCCs). If you have through-ticketing, this most probably does not apply.

• Some airports with several terminals have airside transit between all terminals.

• Some airports may not have airside transit at all between terminals.

• Some airports with several terminals have airside transit between some of the terminals only (for instance CDG has airside transit between all parts of terminals 1 and 2, but does not have airside transit between T3 and the rest of the airport). If you have through-ticketing, the chances of this affecting you are usually slim, but you'll have to check the specific case to be sure.

If you have through-ticketing and you check that the airport indeed has airside transit in your specific situation, you won't be going through Schengen immigration at all. As a US citizen, you don't need airport transit visas either, so the only checks you will go through will be at check-in and at the gate in the UK and at the gate in the connection airport.

Of course, if you have two stops in the Schengen Area, this would require entering the Schengen Area even if through-ticketed.

If you do indeed have separate tickets, you would have to enter the Schengen Area to reclaim your bags and check them back in (before the check-in deadline). In addition to any immigration issues, this opens a whole can of worms when it comes to the time you need, and the risk of missing your connection if the first flight is late (the second airline would consider you a no-show and you would in most cases, especially for cheap tickets, need to rebook and pay for a new ticket, which can be very expensive at the last minute).

A final note: if you are booking a one-way ticket from the UK to the US, consider checking return tickets (with a return two weeks or more after the outbound). Even if you do not use the return leg, it may be a lot cheaper than the one-way alone on the same flight (but a lot less flexible).

What you might be overlooking is booking a ticket on a low cost airline to a US airport (and continuing from there on a different airline if necessary). For example, this is Norwegian from Gatwick to LAX (East Coast can be even cheaper):

of course these prices (in GBP) do not include baggage or anything but still. You could also look at flying to Toronto directly, again from Gatwick. Try skyscanner.net for these kind of searches.