They use the Saros cycle. A repeating pattern that tells you when the next eclipse will happen. Earth doesn't need to be round for that and no round earth maths are used.
Saros Cycle
How would you go about predicting the start of a Saros series?
I don't need to. The Babylonians have already done it for me.
It is a repeating pattern. If I get an eclipse over London ... I know in 6585.3211 days I'm going to get a near identical one. Job done.
Sort of, but not quite.
Solar eclipses are not visible in the same location after one Saros period - they shift by 8 hours, or 120 degrees of longitude, so if your London eclipse was solar, you'd have to travel 8 hours west to see the next one from that cycle. So if you wait three full cycles the eclipse will be back where it started, roughly speaking. Lunar eclipses are different - if the moon remains above the horizon, then a lunar eclipse will be visible from the same location after the next cycle. That's all in the link you provided us with.
There are, however, many active Saros cycles at any one time. This website says it's 40 at present - sounds about right -
https://www.solar-eclipse.info/en/saros/.
The reason that the next
total solar eclipse in London is out of sync with the Saros period (see Modera's post) is because Saros cycles include both partial and total eclipses, so the previous and next total eclipses at the same location aren't necessarily from the same series. The last total London eclipse was from series 145, whereas the next one is from series 155, hence the 91 year gap.
The next problem you have is that cycles are finite - they begin and end. So, if your eclipse in London was the last of a cycle, you won't have one anywhere on the planet from that series in one Saros' time. We therefore need some way of calculating when cycles will begin and end. To make things worse, there is no discernible pattern to them - whilst the period between eclipses in any series is roughly constant, the total number and type of eclipses in each series (partial or total) vary significantly, meaning the frequency of eclipses at any one time varies - unless you know all the active cycles, you can't say when the next eclipse anywhere on earth will be. So you need to be able to predict series starts and ends.
NASA, and other RE believers, use solar and lunar ephemerides derived from orbital models, using various levels of sophistication (the NASA website cites the VSOP87 model for the sun and the ELP-2000/82 for the moon, but plenty of others are out there, all with varying strengths and weaknesses) to predict eclipses and Saros series with remarkable accuracy.
So, my question again: how would you go about calculating the start of a new Saros cycle? Or indeed the end of a current one?