"Britain does not have a CLIMATE. It just has WEATHER"
In hot countries most of the indoor places that need to be comfortable are air-conditioned, and in cold countries most of the places that need to be comfortable are properly heated and insulated. The UK is the exception, since the weather is often interesting and no-one really takes heat and cold seriously. So although the weather can be very cold, very hot, very wet, or just wildly chaotic, it always takes people by surprise.
I think your question should be, "Why is the British weather so odd?" It is something to do with the combination of an almost Arctic latitude (Manchester is the same latitude as Moscow) combined with being an island and having the Gulf Stream bringing warm weather from the tropics on an intermittent basis.
People in the know on geographical meteorology have said that the general situation with the UK weather is even more complex, as many different factors contribute to the changeable weather. As well as the North Atlantic drift, and the topography, there is the fact that the UK is affected by five major air masses: The tropical maritime Gulf Stream, the Polar Maritime air from the North, the Arctic Maritime, the Polar Continental (from Siberia), and the Tropical Continental (from Northern Africa). There's also fast flowing air in Jet Streams (which although shown in weather reports in the USA, are oddly kept secret in UK weather reports). Air goes around in convection currents in Hadley Cells and in Ferrel Cells. Because the British Isles can get weather coming from hot or cold zones, and from land or sea, a changeable hot/cold wet/dry weather is found.