> What's up with the plumbing and heating?

Not sure if this is what GP meant, but Britain has a history of using separate taps for hot and cold water, rather than a single tap that mixes hot and cold to give you a range of temperatures.

Fine if you're filling a basin, not very good for anything else.

> separate taps for hot and cold water

More precisely: boiling hot and ice cold.

Pretty obvious from a UI perspective though (except for the trend of labelling the taps with approx 1 molecule of red or blue ink)

> Doors with a button off to the side to unlock?

It is often power assisted doors that have a button on the side. The button can be difficult to notice, among other reasons because it is low in order to be reachable by wheelchair users.

But is this a typically British thing?

Generally security doors. Button or motion sensor on the inside to unlock it and exit. Probably a keycard on the other side to get in.