"My kettle is rated as 2.5kW, it boils a cup of water reasonably quickly but watch what happens when you turn it on, hot bubbles form around the element, detach and rise to the surface releasing some of their heat on the way up but most of the heat is given up to the atmosphere."
Most of that is dissolved air coming out of solution, and nothing to do with the heating.
"Any water left in the kettle cools quickly due to lack of insulation.
It can't be beyond science to have a 1.25kW kettle boil the same amount of water in a similar timeframe"
As James Joule convinced us many years ago, it takes a fixed amount of *energy* to raise the temperature of water to boiling, so twice the *power* takes half as long - assuming a perfectly insulated kettle, and that's the real point. A crappy uninsulated 1.25kW kettle will be no better than a crappy uninsulated 2.5kW kettle and probably worse because more heat is lost in the longer time to boiling. If you want rules about kettles, make them apply to the insulation, not the heating power.