The marriage I offered was Thomas Robertson’s son marrying the Hart daughter, 20 years after the census. Both families must have kept in touch. It seems to have caused some confusion regarding a second marriage. As for a Birmingham/Robertson marriage there probably was not one.
There could be newspaper articles relating to both families.
I have come across the surname Birmingham before, but a few decades later. They were itinerants in West Wales living alongside travellers, and claimed to be local, with one from Chester, but there was a census that said Ireland. Travellers used aliases and changed their names to suit who they were taking with, so I doubt Birmingham was their birth name. This probably will not help, but those tracing traveller families will fully understand the brick walls.
The name Birmingham may have been used to blend in with the locals, as that area was their intended home. I do not consider it an Irish name. In Wales, travellers adopted Welsh sounding surnames. London was a Guild city, where all tradespeople had to belong to a Guild. Birmingham accepted everyone, which was why many travellers were based in and around that area.
