In both of our London apartments the hot water is a while coming. (The homes before these two were so long ago that I cannot actually remember how the taps behaved there.)
Whenever we turn on the tap, we get only cold water for a good several minutes. If I want to shower, I turn on the water first, then do something else for a while (such as brush my teeth) and then come back for the shower. For quicker things such waiting isn’t worth either the time or the wasted water, so we make do with water that can range from coolish to distinctly cold. Washing the face and hands can be done perfectly well with cool water. Dishes can generally also be washed in cold water without any real trouble, as long as they’re few and not too greasy. (And before you get all anxious about the germs, you should know that the FDA says that washing hands in hot water does not kill bacteria better than washing them in cold water.)
The end result is that I am now so strongly conditioned to expect cool water that anything else feels wrong. When I want to wash my hands after going to the loo at work, or to wash my face in a hotel bathroom, I turn the tap handle well over towards the cold side, in order to get that familiar feeling of cool water. Cool feels refreshing. Warm is for showers, and for thorough scrubbing of really dirty things. Warm water for brushing teeth? Yuck!