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In a previous question on History Stack Exchange, it was reported that the cycle of days-of-the-week have continued uninterrupted since at least 60 AD.

Do we know of any groups of people who, whilst still using the same seven-day-week significantly got out-of-sync with the 'accepted' day-of-the-week?

That is to say a country/island/territory who used the mainstream western/Abrahamic seven day week, but with a different day to the majority - a perfect example might be if between 1243 and 1267, every-day England called Wednesday, France called Monday, so that the countries were always two-days out of sync.

I know that this has happed with the day-of-the-year (e.g. during the adoption of the Gregorian calendar in various countries), but given the importance of the seven-day-cycle to the Christian, Jewish and Muslim religions I would be surprised if this happened in any significant territory, but perhaps may have happened in some small, isolated communities due to a lack of contact.

I would appreciate if any international-date-line shenanigans were not considered.

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    Yeah, I didn't want to include too many requirements in the question, but I would appreciate if any international-date-line shenanigans were not considered. Commented Oct 10, 2023 at 11:29
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    Any of the things that "perhaps may have happened in some small, isolated communities" are, almost by definition,. unknown to history.
    – Brian Z
    Commented Oct 10, 2023 at 12:53
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    It's fiction but too good not to share IMO, IIRC in Defoe's "Robinson Crusoe", Robinson names his native companion "Friday" since he meets him on a Friday 25 years after his initial shipwreck. Upon eventual rescue in his 28th year marooned Crusoe learns that he had lost track of days and Friday was named after the wrong day of the week.
    – AllInOne
    Commented Oct 10, 2023 at 17:04
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    @BrianZ - I was imagining it may be recorded when they reconnected with the majority of society. Something like the first ship to reach Pitcairn Island after twenty years may have recorded that the island was so isolated they had lost tracks of the days and were celebrating Sunday church services on a Tuesday. Commented Oct 11, 2023 at 14:14

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As per the link posted by Brian Z ("One Time Fits All" by Ian R. Bartky), the Pitcairn Islands could have been this at some point between 1790 and 1808; during this period they "lost their daily reckoning" before being given the date by the crew of the Topaz, but there's no indication they were observing any form of daily calendar by then. The Topaz gave them the date on the western (Asian) side of the date line instead of the eastern side, so the Pitcairn Islanders' count was one day ahead until 1814, but as that's date line shenanigans it doesn't count.

However, there is a real and verifiable example of a community observing the seven-day week out of sync with everyone else: the Taiping Heavenly Kingdom. Founded in 1851 as a result of the Taiping Rebellion, at their peak they held several provinces worth of southern China, including Nanjing which became their capital.

In 1852 they instituted a solar calendar, in which months alternated between 31 days and 30 days in length. They also adopted the seven-day week and observed Sabbaths -- but they were one day ahead of everyone else. Quoting from "The Taiping Revolutionary Movement" by Jian Youwen:

This revolutionary calendar, in many ways the most significant cultural achievement of Taiping Tienkuo, nevertheless suffered from two initial miscalculations. The first, still unaccounted for, put the Taiping calendar one day ahead of the Chinese lunar and Western solar calendars, e.g. the Taiping Sunday was Saturday in the Western calendar. This error was never corrected. The second miscalculation, or rather oversight, related to the difference between the actual solar year of 365 1/4 days and the Taiping calendar year of 366 days. No provision was made for the surplus 3/4 day each year with the result that there was a cumulative retardation in the beginning of solar terms amounting to 30 days in the course of 40 years. To allow for periodic correction, Hung Jen-kan years later as premier in Nanking decreed a special short year in every 40 whose months would have only 28 days each.

The calendar never reached year 40; the kingdom fell in 1864, though isolated rebel groups continued to exist for several years afterwards. (Incidentally, the calendar also used the Chinese sexagenary cycle to label days, but that was also one day off.)

More detailed sources are available in Chinese; I found this linked from the Chinese Wikipedia page on the calendar which talks about the discrepancy and its possible causes at length (and suggests that there were rebel armies using the calendar as late as 1869) but I can't vouch for machine translation accuracy.

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