My list of British Gunsmiths and Gunmakers only has one Jackson.

He is Samuel Jackson, 7 & 8 Church Gate, Low Pavement, Nottingham.

This is in the very centre of the old city of Nottingham, not far from the Castle where the Sheriff of Nottingham lived whilst chasing Robin Hood.

Your Proof Marks are from Birmingham 40 miles away, which makes sense.

Good shooting!

HB