i will add a few comments to doc drew's - both sets of proofs were from the birmingham proofhouse, and could be as old as 1887 (when the 13/1 type bore measurements began). skeettx's facebook link shows a fred williams gun that has a much later extensive proofs, all shown on the tubes - so the idea that sleeving removed most proofs has obvious merit. but, there are several remnants of proofs remaining on the barrel flats...so i would bet that the flats were tidied up prior to reproofing. i have an arthur turner/sheffield gun that was reproofed in b'ham under the 1954-89 rules, and it was also struck clean before the reproof - so, such things do exist.
the is the handiest reference for dating british guns, courtesy of diggery haddock -
https://www.vintageguns.co.uk/magazine/514-2from this you can see that the 13/1 & 13 markings could have been done as later as 1953....and the absence of other indicators does not give useful clues to choose between 4 different eras of rules.
but, there is another clue still there - on the water table - ENGLISH MADE. several references given, highlight the fact that williams had a substantial export business, and from memory there was an american law that inported guns required a stamping of their country of origin sometime in the ww1 era, and after 1924 (?) that requirement was changed to "MADE IN GERMANY" or "ENGLISH MADE".
this clue, plus the fairly high serial number would lead me to think late 20's, or perhaps even later.
best regards,
tom