I am delighted to read your comments and find out more about my photo. I wonder if that's where nick have heard 'Vulcan School' before. My family most certainly would not have been attending the private school, so I think the mystery is solved. It is this school for sure.
I think the school may well be the Village School, as looking at a photograph from more recent times (before demolition) the brickwork does seem to match.
it was Coach Road, not Row. I feel like I am getting closer to finding out which school it is. I feel like the brickwork looks similar to the Village School / Vulcan school. If my great great grandfather was 4 in 1891, and I would say he looks between 4 and 6 on this photo, it would suggest it was taken around 1891-1893
Thank you so much for looking into this for me. The little boy on the bottom row, third from the right is my great, great Grandfather David Waddell. On the 1891 census he was 4 years old and was living on Coach Road in Bedlington. I am not 100% sure, but believe that his older sister Mary is on this photograph too. I think she is the girl on the second row down, second in after the little boy standing next to the teacher with the curled under fringe. In 1891 she would have been 12.