We had a problem like this several years ago. Turned out to be a problem with the hot water heater. It was something like defective plastic pipe that had disintegrated - a manufacturing defect that resulted in a class action lawsuit - that, of course, we were too late to persue. Had to replace the water heater and pay a few hundred dollars to rectify the rest of the damage.

This probably isn't your issue, but thought I'd throw it out in case other solutions don't clear up the problem.

The shower head might need to be descaled. Try soaking it in vinegar.

Probably built up sediment in the pipes. Try removing the shower head and cleaning it out.

The shower didn't want to give us any hot water for a few minutes this morning... then it spit out some black specks, and the hot water worked again. No idea what that means. Just moved in a month ago, so have little familiarity with the underlying plumbing.