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We found this 'right' time by phoning TIM. (At that time, the dials of telephones showed letters as well as numbers.)
Phoning TIM was often the task of a young child, as it was something a child could easily manage and would enjoy doing. At the end of the phone a voice would say, "At the third stroke, it will be 11.23 and 6 seconds (or whatever)". This was followed by three beeps. Then the message was repeated as often as one cared to stay on the phone, with the time accordingly being updated. Not that anyone stayed on the phone for long, because telephone calls were expensive in real terms.
Phoning TIM had the advantage that it could be done at a time of one's convenience, but it did cost, and by no means every household was on the phone. (We were for my father's work.)
Another way of finding the right time was from the radio - then called the 'wireless'.
The BBC News began with the strikes of Big Ben as a time-check. (I understand that these were recorded during World War Two in case Big Ben was put out of action in the blitz, as it was important for morale that it continued to be broadcast.) Also various programmes were broadcast at known times, although how accurate this was I don't know.
Setting the clocks to the right time depended on the type of time piece, as explained on the clocks and watches pages - see the side menu.