I think Dave has covered everything quite extensively, but I'll just add my "tuppence worth" that I grew up in homes with no shower until the 1980s. At one time hardware stores, Woolworth, etc. used to carry cheap "shower kits" which consisted of just a simple shower head with a Y-splitter hose and rubber cups which you pushed onto the hot and cold bath (or basin) taps when needed. Their effectiveness was variable at best, particularly in those homes where the cold taps in the bathroom came straight off the mains supply rather than from the cistern in the attic.
Regarding the water chemistry, the west country and some northern parts of England have soft water, but many central areas and almost all of the London/southern/eastern region has hard. It's especially hard water here in East Anglia, which does cause problems.
In this case of the shower though, it seems odd that the unit was working fine until a short while ago and then "suddenly" starting producing barely-warm water. If it were a build-up of scale, I'd have expected some sort of more gradual decline in performance.
