The original Eyot “push-me-pull-you” taps had a hexagon (hexagON – get it?) button at one end of the slide and a round button at the other (round off?) so your one looks like it might be a pattern replacement.
The sealing material in the original Eyot taps was a cork material not rubber although whether natural or man-made I cannot say.
They could be made to 100% seal but were, in the main, better at faintly weeping! There is a modern ball valve replacement that has received praise and unless you are determined to be meticulously original worth considering.
The thread is the standard 1/8” BSP with a nominal 0.383” outside diameter and 28 tpi as Richard says although all the ones I have are 1/8” BSP at both ends. Be aware that there are two types of BSP threads, one is a parallel type (the one you want) and the tapered BSPT type, both have the same thread pitch.
