My first instinct is that if he began to cry when the clock began to strike, that means he was born before midnight. Firstly because the birth comes before the crying, and secondly because the first chimes of the clock would be before the hour. This depends on exactly what's meant by "began to strike", since the twelve notes of the quarter bells might be referred to as chiming rather than striking, but the actual moment of the hour is supposed to be on the first note of the big bell that comes after the quarter bells. (Source: I grew up setting my watch according to the sound of Big Ben, heard on BBC Radio 4 every 6 and 12 o'clock.)
Secondly, days weren't always measured from midnight. Even today, informally, we'd refer to Friday as "today" if we were speaking in the small hours just after midnight of the night between Friday and Saturday. Historically (as well as in different cultures around the world up to this day), the definition of the transition point from one day to the next has varied:
-
Although days are now measured from midnight to midnight, this has not always been so. Astronomers, for instance, from about the 2nd century ce until 1925, counted days from noon to noon.
-- Encyclopedia Britannica
-
For astronomical purposes, Greenwich Mean Time was reckoned until 1925 by the 24-hour clock commencing at noon; since then (in accordance with a hope expressed at the Washington conference) it has been reckoned from midnight, on a notional prime meridian a few metres away from the brass line.
-- Leofranc Holford-Strevens, The History of Time: a Very Short Introduction, Chapter 1: "The Day"
-
Up to late 1805 the Royal Navy used three days: nautical, civil (or "natural"), and astronomical. For example, a nautical day of 10 July, would commence at noon on 9 July civil reckoning and end noon on 10 July civil reckoning, with pm coming before am. The astronomical day of 10 July, would commence at noon of 10 July civil reckoning and ended at noon on 11 July. The astronomical day was brought into use following the introduction of The Nautical Almanac in 1767, and the British Admiralty issued an order ending the use of the nautical day on 11 October 1805.
-- Wikipedia, "Nautical time" (found thanks to our friends at History SE)
Even today, confusion exists about what happens when a baby is born around midnight, and which day should be "officially" considered their birthday. In 1820, in an era of less widespread pedantry and less precise timekeeping, it makes sense that Friday night would be considered as Friday (according to popular parlance) even if a key event took place on the point of midnight.