Is there any airport without customs facilities which has a connection to the UK?
I flew to La Palma Airport. They do have a corridor to check passports for arrivals from outside Schengen, but there are no gates between the luggage belts and the outside. I did not see any customs facilities, but perhaps there exists a system to shepherd passengers and their luggage to a sealed off area that I didn't see, bypassing the luggage belts.
There exist seasonal (charter) flights from the UK to otherwise domestic airports in the northern Nordic countries, such as Kiruna in Sweden and Kittilä in Finland (“Santa Claus flights”). I suppose they must have some way of checking passports but again, at least for Kiruna Airport, I know that luggage arrives in an area that has no separation from the outside world.