We know that Joseph Brazier, the lock maker from Ashes (Wolverhampton so not quite Birmingham) made boxlock actions, including some ffor Greener.
Most makers, even in Birmingham would have bought their action forgings and barrel tubes from specialist suppliers.
My own Edwinson Green best boxlock, undoubtedly hand built to a special order, was ordered with a “Scott club head action” and Whitworth chopper lump barrels.
By way of exception there is a fascinating article in the latest Historical Breechloading Small Arms Association Journal illustrating the wooden patterns used by a craftsman at Purdey who used to hand forge action bodies for such oddities as a .410 over and under.