Edwinson Green were a very good gunmaking firm. They also made the guns for London Sporting Park.
At various times a number of the provisional makers would have had time served craftsmen working for them or as local to them 'home workers', as we might call them these days. So I fear that you would be unlikely to be able to come up with a definitive list. The era has a bearing, too.
During the depressed years of the 1920s many provisional guns makers, having lost trained men during WW I, were reduced to buying in their guns from Birmingham and merely putting their name to the gun. They would have carried a small stock of ready made guns to sell on spec to folk visiting their shop. They would have had a small staff of gun smiths, able to carry out the usual repairs, or they would have sent the gun off back to Birmingham for more complex repairs.
However, if we look at the hey day of 1880-1915, I dare say you would find more provincial makers able to complete most of a decent game gun. They may have bought in the locks, action, barrels and ejector work, ready to hand fit together, stock and finish, or in a few cases may even have been able to do that work themselves. Examples would be Gallyons, Elderkins, Rosson
Going further back in time, to the percussion and flintlock eras, I suspect there were many more firms in the provinces making sporting guns and pistols in what was then an unregulated business sector. Examples would be Turner, Richards
Tim