Good point Thunderbox, but on the basis that Woolwich Base Workshop was the same as, say, Catterick or Aldershot or Warminster, then they wouldn't have access through the stores system of the 'master component', in this case, the body as this part was never available through the Ordnance system. The strict non-availability of the master component prevented the manufacture of small arms by anyone. If they were, you could end up with a situation like the unnumbered/unaccountable LB rifles. Standard No8 rifles were available to any unit. Maybe the W serial number No8's deserve a bit more investigation but I doubt that they were made up by a REME workshop for reasons I've given. Mind you, there could be quite a few out there if we've got number 4 and 53 already!
I can't see any unit being allocated 50+ No8's for any reason so that would eliminate all Army units as they'd be allocated from Ordnance in any case. Even the big training regiments that I can recall would only have 12 or so. I think that they were allocated for a project. Clearly there is no previous designation or number under the W mark.
The big Base Workshops would manage units 'private' stockholdings (we called it the PRI stores), such as shooting team No4's and especially Army/Navy/RAF Cadet forces who had 'private'/usually ex WD weapons such as my sons old school who had 6 or so telescope-less No4T's that were used as accurate target rifles with PH5C sights by the shooting team............. But I'm going on a bit!