Funnily enough, I did think that maybe I'd obscured the slot (my first attempt I didn't realise the slot was a thing - I've not done a basin before!) but part of my second attempt was checking it was all clear and lined up correctly. I'd had the idea that if the overflow couldn't allow air in, then it would perhaps make a vacuum, which would stop it draining.
I am not sure about the pipework which leads to the drain. It's a 1930s semi, where the WC and bathroom were combined almost 20 years ago so much was altered. I don't know if it goes into the soil stack or via a hopper. The waste pipework under the basin (in a vanity cabinet, of course!) is solvent-welded... there's a 90, it goes down, but I don't know where, nor how it related to the bath or shower waste, as it's the opposite corner of the room.
I think I'll have to try pouring water directly into the drain (tricky without making a mess!). Then try with both traps (minus basin). Also test the traps just by filling via the bath tap (again). Hopefully that shed some light.
I'll get a couple of basin mates and maybe see if I can get some spare washers too.