The year used to run on the quarter days - see my post above. You are right about some taxes in the UK - I think the year for corporation tax starts on 1st April. As usual for Britain, it’s a bit messy.
For one thing, there aren’t good eclipse records going that far back, and for another, the Babylonians did not use the regular immutable seven-weekday cycle that was later popularized in many cultures by Hellenistic astrology as well as by the spread of Christianity.
There do exist some records of eclipses from medieval Islamic astronomy (ninth c. CE onward) that mention the weekday, and they almost all agree with the weekday that modern astronomy reconstructs for the eclipse in question. The occasional discrepancies could arguably be due to authors’ or observers’ errors rather than to any systematic mismatch between medieval and modern weekday counts.
So yeah, the current seven-day week cycle has apparently been functioning without a hitch over large parts of the inhabited globe at least for a dozen centuries or more, although it probably doesn’t stretch back uninterrupted for longer than two millennia, if that much.
The weekday cycle is an example of knowledge made robust by massive redundancy, i.e., almost universal public awareness. More complicated intercalated or otherwise astronomically adjusted calendars have often suffered discontinuities over the course of their history, because only a few experts in any culture have ever been able to monitor and maintain such calendars with full understanding.
But once a simple weekday cycle gets going in a culture, it’s very resistant to systematic error. You as an individual may forget what day of the week it is from time to time, but there will always be lots of people who can remind you!
Quite a number of North American firms use Mar 31 as year-end for accounting too. Of course, legally you can use any date (as long as it’s consistent), and there are a few that use some other month.
As for contiguous days of the week - it’s hard to imagine any catastrophe or disruption that would disrupt the entire Jewish/Roman/Christian world sufficiently to cause the whole area to forget the day count they have in common - except maybe for Israel BC, for example the Babylonian captivity. (600BC?) At that time only they would be on that 7-day cycle.
After all, if the invasion of the Huns or the Mongols made a whole country forget what day it was, well, the rest of Europe and the Middle East would likely not; and the monasteries were all about remembering and preserving the week cycle.
Also, we tend to forget the power of the human mind. With writing, we forget things if they aren’t written down; with computers, we forget time and details; with TV to keep us busy and reference books to consult, almost nobody spends their nights just looking up at the stars. I know if I’m on vacation and off my routine for more than a week I start to forget what day of the week it is. However, way back when, people would remember these details because they had to - there were no watches with date function or similar reminders.