Doesn't help with a baptism, however there is this death notice:
"At Haydon Bridge, on the 3rd inst., aged 56, Mr. Robert Davison, stone mason."
Saturday, Aug. 11, 1849
Publication: Newcastle Guardian and Tyne Mercury
Also a marriage of Robert DAVISON, mason of Haydon Bridge to Mary FOSTER of Hexham on 17 May 1840 - another possible son of Robert? Though there certainly do seem to be a lot of masons named DAVISON at Haydon Bridge.