I thought surely the numeral zero had been invented in Roman times, it just hadn't made its way to Europe yet?

The Khmers had it in 10683 HE:

https://www.smithsonianmag.com/history/origin-number-zero-180953392/

The comments are angry but informative. The claim that this is the earliest record of "our" zero seems iffy, Arabs seem to have been quoting Indians as using a zero earlier. But that's the zero with a lineage leading to Western Arabic numerals. What about the Mayans?

I suspect that analog clocks as we know them also postdate the arrival of zero even in western europe, depending on what counts as an analog clock and what date you consider as “arrival” between the first uses (11th century) and widespread adoption among most of the population (as late as 16th century).

And I can't find when Europe moved from the old system of dividing the day and night into 12 hours each of different duration (and possibly only bothering about dividing the night in just 4 vigils) and the modern systems of hours of constant length, starting at midnight, but I believe that came *after* mechanical clocks, possibly even a few centuries later, when zero was pretty much a thing.

(I forgot to add, near to analog clock: e.g. the ones made possible by https://en.wikipedia.org/wiki/Verge_and_foliot mechanisms, from the 13th century)

WRT zero, using a positional system and having zero as a number are pretty different things: the former is just a convenient notation, while the latter means recognizing zero as something that you can do operations on like any other number such as 1 or 2.

Among the cultures I know of, zero as a number appeared first in India in the 7th century (https://en.wikipedia.org/wiki/Brahmagupta).

In greek mathematics, where numbers were mostly considered as ratios of segment lengths, the concept of zero had no meaning, and thus greek-inspired math in europe (and in the islamic world) had a hard time recognising it (and negative numbers) as valid numbers and not just something useful for calculations, even after they had been using the positional system from India for a few centuries.