On Saturday, November 7, 2020 at 8:35:52 PM UTC, Dave Liquorice wrote:

Normally splines rather than screw on, though there will be screw under a cover to hold the handle on.
Sorry, poor explanation on my part €“ the plastic tap head pushes on (no screw). I was referring to the €˜thread inside the plastic tap head that the splines fit into, which is now worn.

Splines look a bit knackered but a new handle ought to fit and the possibly bad assumption that they haven't changed the number of "standard" splines theer are over the years...
Ive tried swopping the cold and hot around. The €˜good plastic tap head from the cold works on this hot top, which suggests the valve & splines while ancient, still work.

Ive also tried swapping a valve in from another sink and that works.

That's a composite kit with various collars and washers to enable the valve body fit 1/2" of 3/4" taps. At aguess you'll need the right hand collar and the smaller conical washer. You may need to remove the flat washer to fit the conical one. Not sure what open top hat things are for. Fiddle about and see where they fit?
The 2nd from the left on the middle row works to allow me to screw the valve in to the tap itself.

The problem I then have is that the new valve is too short to turn the flow off. Im pretty sure it needs one of the two fittings on the left of the front row, which I assume fit over the bottom of the valve. The problem is that theyre a hard plastic and I cant get them on. Ive tried warming them in water, but no joy.