The power of the kings has fluctuated with time, and there have been several low points at which the king was weak. When Henry II killed Archbishop Thomas Becket in 1170, the Church had him publicly whipped in Canterbury Cathedral, showing who was in charge in that period. Oliver Cromwell executed king Charles I in 1649 and ruled for 9 years as Lord Protector of England, unquestionably the most powerful figure. Soon after in the Glorious Revolution of 1688, Parliament dumped James II with little protest on suspicion of excessive Catholicism. The insanity of George III in his later years led to the Regency, in which the king was not in charge of anything. The final blow was probably the long reign of Victoria, who pragmatically acknowledged the figurehead status of the monarchy and cemented it from 1837 (at which time her office might have had more sway if held by an ambitious monarch) to 1901 (when the monarchy was just as we would recognize it today).

Aside from the execution of Charles I, which is really about a lot of factors, I'd say that the transition of power from king to government/prime minister occurred during the reign of George I, the first king from the House of Hanover (who hardly spoke any English, being born in Hanover). One of the key factors in this slow transition was the sheer level of statesmanship displayed by Robert Walpole, the country's de facto first prime minister.