Any water coming out of the cold tap comes usually direct from the mains without visiting a tank anywhere on the premises.
I think you will find in the UK only the cold water tap in the kitchen is direct from the mains, traditionally the bath and wash hand basin cold feed was from the attic tank.( with all the associated contamination)
Some modern plumbing solutions are direct feed cold and mains pressure hot with a small header tank to flush toilets, but they are not usual, most plumbers don't bother with attic tanks for flushing toilets if fitting combi boilers, and connect direct to the main feed.