Now then. Eastgate subway.
All indications I've been able to find seem to be that the entrance to this subway is behind the large black wooden hoarding on Lady Lane, opposite the bottom of Templar Lane:
https://goo.gl/maps/kPXyJih2DYw73hDs7
A while ago I asked around some colleagues in the council and they confirmed that is where it is. At the time Victoria Gate was being built on the other side of Eastgate, there were discussions about the possible subway, but it wasn't fully explored as there was no need to expand the span as far as this side.
This photo from 1956 (a year on from the initial photo that Phill found at the beginning of this thread) shows the buildings completed, albeit with scaffolding up, and the subway entrance visible:
https://www.leodis.net/ViewImage/ByIndicator/5436
The description of that original 1955 photo at
https://www.leodis.net/ViewImage/ByIndicator/5533 has also been updated:
"This image, showing a man holding a plank of wood on top of scaffolding, a bucket in the foreground and a corrugated iron fence behind the scaffolding, was said to show Eastgate subway under construction, but we have found no evidence to confirm the existence of a subway under Eastgate. According to annual reports of the Leeds City Council’s Town Planning and Improvements Committee between 1956 and 1958, a block of buildings was built on Eastgate which incorporated the entrance to a proposed vehicular subway running under Eastgate to a proposed underground car park. However, there is no further mention of this vehicular subway in later reports, so it may be that it was never built and the entrance just led to car parking accommodation in the basement of the buildings."
This fits with the fact that there was no sign of a corresponding subway entrance/exit where you'd expect to find one - ie, at the rear of the former buildings on the other side of Eastgate (Union Street as-was, rear of Hoagy's etc). It sounds like the plan was actually for it to provide access to an underground space below Eastgate, rather than a thoroughfare type subway. There is another basement car park further up Eastgate / Lady Lane.
The odd thing about the idea of it being a car park entrance, though, is the three rather large concrete pillars. These would seem rather awkward to drive a car through, although cars were smaller back then I guess!
I walk up Lady Lane several times a week and it is possible to peep slightly round the edge of the hoarding, although I've not been able to glean much - just that it extends a long way beneath the building.
Possibly a bit of a naughty suggestion, but does anyone who has a drone camera fancy sending it over the top of the hoardings on a little surveillance mission?