Gene,
At the risk of differing with WWG, it all depends on what you mean by “London” and what you mean by “barrel maker”.
The Royal Small Arms Factory at Enfield is within the London area, and I am sure that they made their own barrels for Enfield rifles.
What I don’t know is whether the iron or steel came to them in rods, billets or rough bored tubes.
Cogswell and Harrison manufactured shotguns in London, and London Small Arms produced Lee-Enfield and War Office Pattern Rifles, but again the same caveat applies.
By “gun-barrel welder” I believe WWG meant to refer to someone who could produce a gun barrel with no greater industrial means than an anvil, furnace, hammers and a mandrel. W.Fullard appears to have been the last such practitioner in London.