I've done a bit of research since posting this. Here are my findings so far, it sort of answers the question but not completely. If anyone has any other information on the history of the water supply in the UK it would be much appreciated.

In 1942, Winston Churchill came into contact with a single tap system and found it to his liking. This means the system definitely pre-dated the war, which was expected. Source.

During a wartime visit to Moscow in 1942, Winston S. Churchill discovered a marvel of modern technology: hot and cold water flowing from the same faucet.

The plumbing in the villa where he stayed as a guest of Stalin was unlike the primitive British standard of separate taps for hot and cold. Rather than having to fill up the sink to achieve the right blend, the British leader could wash his hands under gushing water "mingled to exactly the temperature one desired," as he put it in his memoirs. From then on, he resolved to use this method whenever possible.

This website states that the British system pre-dated the invention of water mixing valves. I can not find the second source cited in the article online.

The UK started to build the public water systems in the early 1800 [1]. Taps in those days did not have the non-return valve. The mixer tap was patented only in 1880 [2] ... I guess they were expensive at first. And the non-return valve was patented in 1907.

As a note, the first source in the above article gives some interesting history on the water supply system in the US.