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He is listed as "Prince Regent" in Taboo. So yes, he is in it but not sure if that is the king or not.
Mark Gattis, who is also the current show runner and the person who brought back the new Dr. Who, produces Sherlock and plays Mycroft, is playing the Prince Regent.
The Prince Regent is what we think of as King George IV. The deal is that George III ruled as the regent, not as the monarch. His father, King Geroge III was still the monarch but had been stripped of the power to rule. Because George III was mentally ill, his son took over as Prince Regent, the de facto king. They probably don't bother to explain this in the show because it was originally a BBC show and people in Britan probably all know this stuff.