James provided the following:
There is a book; The North British Railway in Northumberland by G.W.M. Sewell ISBN 0-86303-613-9 that covers the history of this railway. Unfortunately it does not mention the date of demolition of the Signal Box but the Border Counties Railway bridge here was demolished (with difficulty – the first crane fell in the river) in October 1959. The original signal box of circa 1839 was built on the north side of the line but erosion by the river led to it being moved to the south of the line in 1867/8 when the Allendale branch was opened.
There is also an image of the bridge in the HLHS Photo Archive, Hexham, Miscellaneous buildings/Structures.