It surprises me, frankly, you get to the 19th Century.
Almost all of the know-how involved in making basic steam engines - even fairly high-pressure ones - and building most early Victorian infrastructure is artisanal. You need someone with an understanding of basic physics to figure it out, but the maths required to do that are fairly simple. Once you have it figured out, you vcan teach it hands-on. Of course all that technology will be built with big margins of safety (and still be dangerous), but so much mid-19th century tech was. Did you see the bridges they built for the transcontinental railway in the 1860s? Like that.
The main problem, of getting to the nineteenth century IMO is that so many of the things that motivated technological development depended on modern maths. The technology itself, not so much. Remember that well into the second half of the century, many engineers were self-taught, often with only basic school education but a wealth of practical knowledge. You didn't need to be ablke to solve for x to build, maintain or drive a train. The problem ism, you needed to be able to do that to manage a railway line.
This, I'm less sure of. AIUI, the tally stick stopped being able to cope with increasingly sophisticated (complicated) businesses, whence (more/less) modern accounting. Am I wrong this needs zero?
You can have a good deal of functioning accounting with fractions and non-zero numerals. If you look at papyrological evidence from Ptolemaic Egypt, you see stuiff that sounds susapiciously like modern tax consulting. But there is a limit, both in the practicality of the system and in its ability to model things. I'm not an expert by any stretch of the imagination, but I know that without decimal points and the operations they allow, adding up, dividing and multiplying fractions is major PITA already. Designing a mathematical model of even the simplest economic transaction becomes horribly difficult.
THink of it by the nanlopgy of building high structures. You can do surprising much with cut stone. The pyramid of Khufu was the tallerst man-made structure on earth for ages. But to get even a little higher requires an enormous amount of additional labour, which may well explain why people stopped building taller pyramids. If you have a system that solves your basic problem more elegantly - like arch vaulting, steel beams, or reinforced concrete tubes - height can be added with much less difficulty. A zero system give you that ability. Once you have it, you can do algebra, and with that, you can build stuff like space shuttles, nuclear reactors and structured debt obligations. Without it, your ability to track and model economics, populations, and observed phenomena stays fairly basic.
That's about what I figured. Post-classical physics generally? Genetic engineering?
Very unlikely IMO. I doubt you could get a working Newtonian model. Elements of it, maybe, and geometric approximations of the thing. Certainly no stochastics. I suspect almost anything to do with modern chemistry is out.