Would have thought there'd be a few with it being a port town, yet there's only one listed on Yell.com oddly:
Game Fair Shooting & Fishing
12, Marygate, Berwick-Upon-Tweed, Northumberland TD15 1BN
Tel: 01289 305119
had a bit google around:
R Welsh > Fishing Tackle in Berwick Upon Tweed
Tel: 01361 883466 - 28 Castle Street Duns, Berwick Upon Tweed, TD11 3DP