I have a 1968 home with a shower(walk-in) (in master bedroom not only shower in home) of the same age. It is tile (lavender and white) and the tile is starting to buckle on the back wall... wall on opposite of shower head. I believe it is time to repair and so a shower insert was my next thought. I am sure not 100% that it does not have green board behind it but I bet not. The other thing is that it is only 32" wide most inserts are 36"+. How would you describe the way to best tackle this job? PS just had local plumer put in NEW shower head pipes behind wall and facet knob. so that is like mmmmmmmmmm2 months old.
Sounds like aome major demo is in your future. You might be able to cover it with something like Corian or Swanstone, but i would not recommend those thim plastic surrounds. If i were doing it, I would tear it out to the studs and start over. Either with tile or a man made stone base and surround. No need to disturb that new faucet.