Hi and welcome to Birmingham History Forum. There may be other members on the forum who will know more than me about it history. Looking at it on Goolgle Maps its quite an attractive building that I imagine was built about the same time as the social housing adjacent to it.
The government have a strict criteria for listing the building must meet. For example, Grade II buildings are of special interest, warranting every effort to preserve them.
Here is a link to the governments website for guidance.
I don’t think the Victorian society would be interested but
Save Britan’s Heritage (S.A.V.E) may. They are quite helpful and have a lot of knowledge.
Most certainly to save a building you have to have a place that tells a story, so researching its history is a good way to go. Have a look on face book or Instagram if you like that sort of thing, there may already be a group, and social media can be quite a powerful too.
Maybe looking at MyCommunity to see how nominate an Asset of Community Value (ACV).
Most certainly the way forward is to help people to take responsibility for the local buildings, artifacts and landscapes they love and help give them the power to transform people, places, and communities.
Do come back if you need any further help