Manchester isn't quite my patch, but I'll have a go.
National Library of Scotland are gradually digitising their OS map collection
Christ Church, 1890s -
here - seems to be about where School of the Resurrection is now.
You can opt for the 'side by side' option to see it next to current maps, and / or other era maps (looks like they haven't got the post-1945 very large scale on there)
I've had a quick look but can't see Spencer Street. Church parish boundaries may not match the borough ward boundaries shown on the map.
Basic google searching hasn't got me very far, as I can get references to various Spencer Streets, none of them seem to be the right one. And there being another Bradford doesn't help.
bit about the church
here.
this has some pictures of the general area before redevelopment, but no mention of spencer street.
manchester university also have a map collection online -
here
manchester archives website
here - they are likely to have old street directories of Manchester which may help.
Leicester University have digitised many of them and may be worth a look.