Possibly not Birmingham, but how about Britannia Iron Works? Most of the time if they made/peddled bicycles & sewing machines, they also were involved with guns(nothing definite just yet) - a little history:
http://www.nelmes.fsnet.co.uk/paxman/paxbrit.htm Or along/near the Birmingham canal, which allowed commerce, was a Bradley furnace & ironworks which had been making steel for sometime using Darby's discovery. And a link for that info at:
http://www.bcn-society.co.uk/bp167_rayshill.php which may have been a source for tube material early on.
And there was Bersham Ironworks which was destroyed by Wilkerson:
http://en.wikipedia.org/wiki/Bersham_Ironworks who ran the Bradley ironworks.
Kind Regards,
Raimey
rse