I would probably just get a pre made acrylic or fiberglass kit - they usually come in a couple of pieces and are RTA. They do the job and have been around for quite some time. It'd probably be harder to source your own fiberglass panels and fabricate them than it would be for the minimal savings.

I keep looking for something like that but they all seem to require that they go into studded out walls. The Mustee ones are completely free standing and don't require building out the space to install. Perhaps I am not searching correctly?

Yeah, I did some googling and it seems like that brand really has the market cornered, at least in the US.

What is your goal in improving the shower? If simply to lessen the ick factor at a low cost, I would buy a new replacement unit. Even if you built a box with treated lumber, put up some fiberglass panels and sealed all the seams, it still seems like the cost/benefit is comparable to just buying a new mustee unit. The only cheap fix I could think of is doing something like spraying it with Krylon Fusion paint to liven up the surface. But you would have to take some time to prep it.

If you're overall trying to start improving the basement space, get yourself some nice straight 2x4s and a framing gun and having a go to frame out a place to put the shower.

Did you ever find a solution? Five years later Mustee is all that comes up for me too in my search for a stand alone shower!