] I believe it was only with the arrival of railways that there was any uniform British time.

Same is true in the USA, where it was finally instituted by federal statute after the Civil War. Uniform time over a large area became necessary, in lieu of local solar time, only because of railroad speed and the complexity of railroad scheduling. Timetables had to be expressed in a consistent standard not just for the convenience of passengers but because an ambiguity or inaccuracy of a minute or two might mean collision. In the later part of the 19th century, advertisers would boast of their watches that they were of "railroad accuracy." Official station clocks were automatically set once a day by telegraph.