We have a small washbasin in the downstairs loo which has a bottle rather
than U or P trap. It has strange connection on the downstream side of the
trap: https://s9.postimg.cc/i655dw0kv/bottle-trap.jpg - it's got a
half-round connection into the outlet pipe of the trap, with a threaded
connection (smaller than standard 1.25") which is covered with a blanking
plate.

I'm intrigued what it is intended for. My first thought was that it was for
an overflow to be connected if the basin did not have its own internal
overflow connection that joins the normal output just below the plughole.
But since the connection is on the downstream (smelly) side of the trap, any
sewer gases would vent through an overflow (or anything else) connected to
it.