In the 1900 Kelly's Directory, 37 Bailgate was the Barley Mow public house kept by Edward Winning. It wasn't a pub in the 1960s or I would remember it.
It was on the west side of Bailgate, about halfway between Chapel Lane and Westgate.
A current directory shows 35 Bailgate, Lincoln, Lincolnshire LN1 3AP is Bailgate News, tel 01522 526330. You could always call & ask whats next door!?!
Of course, it's always possible that it was renumbered, but in the 1933 Directory, 37 Bailgate was a chippy - one of two owned by Walter DERRY, fried fish dealer.