Winfer's book will give you all the detail you'll need. I believe this is what he referes to as the 3 B Action (pg.36-37). The three screws in the side of the action identify it as the # B Action. Winfer's book makes no mention of J Burrow.
... but in Nigel Brown's British Gunmakers Vol.2, there is a short entry that reads, "JAMES BURROW LTD., 117 Fishergate c.1917 (pg.184). This date would agree with the nitro proofs on the barrel.
I agree with the comments above. You have a fantastic rifle in original condition. As for British single shot rifles, "It starts with an interest, turns to passion, then migrates to obsession” . Congratulations ...