Found a couple of photos with J. E. BARNES shop at the Market Place and I think that is the Post Office in the top photo, left of J E Barnes as you look at the photo. Don't know the year these two photos were take.
I spotted a postcard of the Market Place that John Ashford had posted on the Bygone Bedlington group and he was asking if any member could date it. All the estimates for the year were late 1950's to early 1960's. Enlarged the postcard and there was the Post Office next to Barnes :-