It is highly unlikely such a tunnel exists, the church was built around 1900 so any tunnel would not have been built earlier than this, there was also considerable difficulty in raising funds to build the church.
As it would be substantially under roads the council would have needed to approve such a structure in that era, additionally it would be going underneath a number of other buildings unless it veered considerably from its near 100m direct line.
I can't immediately see any records of quarrying in that area so the sandstone is probably not of good enough quality for tunnelling safely. A quarry that close to the docks/river would be quite desirable. Nor are there any records of landfill usage which could be taken as an indication of quarrying.
It sounds typical of one of the myths generated by one of the local ghost creating historians.