The great Robson centre in the area was Stamfordham parish, in particular Ingoe, a farming settlement. A branch of my family came from Lionel (originally from Ingoe, according to the Bishop's Transcript) and Jane Robson who married in 1784 and settled in Great Whittington, part of Corbridge/Halton parish, next door to St. John Lee. They were having children up to about 1803, so are unlikely to be Walter's parents, but their oldest sons - Joseph, William, Lionel-would be be the right sort of age. Of cause Lionel senior has brothers who also stayed in the area.However, I've had a quick look at my notes from the earlier Corbridge and Halton registers and can't find any Walters among the Robsons. Names did run in families, so that could be significant. What did Walter call his eldest son? That could give a clue to his own father's name.
Interesting that you can't find a baptism for St John Lee for 1813, given my earlier post about not finding a baptism for John Little in 1812 for that parish.