Spoke to my Dad who's 87 who remembers a few tunnels or similar structures in different parts of Bedlington.
The Church Lane tunnel could possibly be the culvert where Jocker Amos was found following the Sun Inn murders. There are some images of the culvert elsewhere on this site.
There was a surface drift which was part of the Doctor Pit which ran under where Tescos is situated and started nearer to the gas works/police station.
There was also a 'cundy' which is a type of underpass which ran from the colliery rows to the gas house near the new police station bypassing the Doctor Pit yards.
There are also 2 tunnels which emerge at the bottom of the Furnace Bank right on the river bank next to where the bandstand used to be. I think one of them is clearly visible although the entrance is silted up. This one may be the one with 'abide ye' or similar carved on the stone arch. If you're looking for it, it's about 50 metres west of the Furnace Bridge on the north side. Dad reckons these structures were actually water inlets which filled up at high tide. Several bore holes were sunk into them along their length to obtain water for the ironworks.