Round-the-clock means it does not stop when night comes: the care, the shop, the news desk, the watch on the river — it goes on through all the hours, day and night. The picture is the face of a clock: you travel all the way round it, hour after hour. British English likes round-the-clock as a hyphenated adjective before a noun (round-the-clock care, a round-the-clock service). American English often writes around-the-clock, same meaning; you will also hear around the clock as a phrase after the verb (they worked around the clock). It is ordinary, slightly news-like English — not slang and not stiff. Though actually, people stretch it. It does not have to mean every single minute with no rest, in a scientific way; it often just means a very full, almost non-stop service. A hospital can truthfully offer round-the-clock care; a café that stays open ‘late’ is stretching the word if it still closes at midnight.