Sounds like the main drain for the bathroom may be restricted.
As you shower the water doesn't drain away fast enough, and the plumbing begins to fill with water. As the pipes fill, air may be forced out through the toilet trap causing "bubbles". Since the tub is the lowest drain, the water backs up there first. If you continued to shower, you'd eventually see the toilet overflow as well (if the tub edge is higher than the toilet rim). If the tub was deep enough, you'd eventually see the sink overflow too.
If you look at this crude representation of your bathroom, you'll be able to see what I mean.
Now, if we add in a clog we can start to see what happens when you shower.
As you shower the water cannot pass through the clog fast enough, and the pipes begin to fill.
At this point, any air trapped in the line between the drain and the toilet trap will be forced out and bubble up trough the trap.
As you continue to shower, the tub is the first to back up.
Eventually, if you shower long enough the toilet could also overflow.
Depending on the height of the sink, quite a bit of the bathroom would have to fill before it overflowed.
The first thing I'd try is to clear the drain using a Plumber's snake. A hand auger would likely work, and can be purchased at a local hardware store fairly cheaply. Drill powered augers are also available for a bit higher price, but require a bit more of a gentle touch (and obviously a drill).
Another option is that the vent for the bathroom is clogged, or not sized appropriately. If vent stacks are not covered properly, leaves and other debris can become lodged in the pipe and prevent air flow. A clogged vent will cause the fixtures it's venting to drain slowly, and the results can be similar to a clog as described above.
As with the clog, the solution to clear the blocked vent is a Plumber's snake.
There is no secret to getting to a cement shower. I just took one out of my master bath and if I didn't have a lot of reasons for doing it then I would have left it. They are great if done right and you can basically retile them infinitely...
So first you should try to get to your valve from the side or back. If that won't work you will have have to cut out a piece of the cement. I would in no way take out the whole shower. Just a cutout where you need to reach the plumbing.
Next you have to get rid of all the tile to figure out what condition concrete is in. Most of the time (95%) it is in great shape. It may need to be slightly patched or smoothed out then waterproofed. This is very easy. Really the only thing that go "bad" with these showers is the floor. Sometimes you will have to make a new base because of extensive water damage - this can be a big project.
But there is nothing wrong with a concrete shower. And the mold, grime, and mildew is probably a case of poor tile install or old tile not being grouted as you have said. If things are in good condition under just retile. If the pan is in good condition then this is a no brainer. Post pictures if you are thinking about demoing the whole thing (might be talking about couple thousand pounds of cement for a big shower).
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Yes use their descaler they recommend it for a reason. Life expectancy is 10 to 20 years. It is out of warranty by 14 years and close to end of service life. Take care of your poo squisher.