I tried unscrewing the dome section by hand yesterday (presumably anticlockwise), no luck, presumably I’ll need some kind of wrench.
I’d probably need to isolate the water at the stopcock as no local isolator just copper pipe directly to the tap fitting.
The bathroom needs updating in all honesty, so would rather not spend on new taps unless essential currently, though appreciate I could mess it up! The taps have worked for decades and I suspect it’s a fairly trivial fix.
I spent until late evening gardening there and had a bath, but it was too late to find tools in the garage to fiddle further.
I had a similar issue with a very similar tap a few years ago. It confounded all my efforts and even prompted me to purchase some rather specialised tools, all to no avail. In the end I had to admit defeat and called in a plumber. That was the best thing I did and something I should have done immediately. Half an hour’s work and the tap was back to working perfectly - all cleaned up inside with new washer fitted (the old one was an old leather washer!). £60 well spent.
1 What fatcat said, except I’d suggest first spraying it with some sort of light agent like WD-40 and let this soak in for a while.
2 wrap a bit of rubber around it like a rubber band to protect the surface
3 apply a mole wrench, not too much pressure though or you’ll compress the dome. With a bit of luck it will ease undone.