Yes, people do wear airline PJs in business class. Not all, but some.

I usually wear my BA First or QF First or Business PJs (as BA does not provide PJs in Club World). I probably wouldn't wear my own PJs, but would perhaps change into something comfier (such as tracksuit trousers). I tend to change after dinner and before breakfast.

I find the washrooms adequate enough to change in, but if the cabin is dark and the divider is up and you are sitting in a middle/window seat, changing in the CW seat is possible if done discreetly.