Most of the time and in most areas it is perfectly safe. I've drunk it all my life and so has nearly everyone I know, with no ill effects at all. The Drinking Water Inspectorate checks the water supply regularly and finds it safe; tests have also been carried out where people didn't know whether they were drinking tap or bottled water, and in the Uk most of them couldn't tell the difference; so taste isn't an issue for most people either.
In many parts of the country the water is fluoridated; some people think this alters the taste (though again, the results of a blind test would be interesting) but it certainly doesn't affect the safety, in fact fluoride is added to water because it's good for your teeth.
There have been occasional problems with the water supply, but these are rare and well documented. In short, in the Uk anyone who tries to sell you bottled water is just after your money, and there are much better things to spend it on.