Take a scrap of wood, trim it square to a size slightly larger than the core of the threaded hole, taper the end slightly to provide a lead and screw it into the hole as far as it will gp. There is nothing very fussy about all this and if one dosent work then a scrap of wood is cheap enough!
With any luck when the wood is unscrewed there will be a clear impression of the thread in the wood which can be measured to find the pitch. Once the pitch is known that will narrow the search for the right thread.
For reference 1/8″ BSP has an O.D. of 0.383″ — core 0.337″ — pitch 0.0357″ — T.P.I. = 28
Near threads are:-
3/8″ BSF — O.D. = 0.375″ — core = 0.311″ — pitch = 0,050″ — T.P.I. = 20
3/8″ UNF — O.D. = 0.375″ — core = 0.399″ — pitch = 0.0417″ — T.P.I. = 24
Metric, 10mm is a bit small, 12mm a bit big, there are 11mm (0.433″ O.D.) threads and also a 1mm (0.040″) pitch series.
These are the common ones but there are more obscure ones too, if you could post the pitch I can dig deeper.