Probably the Lizard here in Cornwall for southerly mainland - there used to be a little gift/pasty shop, not sure if anyone lives above it. Maybe there are still manned lighthouses that are further south!
Quite likely the Youth Hostel at Lizard. It is on the most southerly promentary, actually nearer the sea than the lighthouse, and its the garden wall is at the top of the cliffs. If the postman attempts to got any further south, he'll get very wet.
Looking at the OS map of the north of Shetland, the most northerly building looks to be in or near the settlement of Skaw on the NE tip of Unst, which is nearly 60.5 degrees north. If the lighthouse on Muckle Flugga is manned, I suppose that could have a postcode too and that is as far north as the UK goes!