I recently explained this on a Memory Alpha edit, but they took it down. Apparently it was speculation or some such and MA is NOT about that, however logical the speculation may be, or how well based on known information it is.
But they did, at least once, convert military time to standard time in the DS9 series. Somebody mentioned to Odo something was to occur at 21:00 hours, and told him that was 9:00 p.m, as if he didn't know how to convert. (IIRC, those were the numbers, but don't quote me, and I won't hunt for the episode title, either, but I recently saw it). And Odo replied, he "knew" that, meaning he could convert military time to standard non-military time. This proves they were using a 24-hour clock there and not a 26-hour clock. Otherwise, 21:00 converts to 8:00 p.m. by subtracting 13 hours (half a day) and not 12 hours.
Of course, one may also use a 26-hour clock at times and have two systems of military time. Which one is being used at any given time is likely discernable from context. Any mention of a time past 24:00 to 26:00 would obviously be using a 26-hour clock.
However, I further explained that many (if not all) rotating worlds likely divide their day up into 24 equal parts since 24 has many handy factors, just as they might use a dozen (12) or 360 degrees in a circle, as these are not arbitrary or randomly chosen numbers, but chosen for their many and convenient factors. 24 units, for example, has 1, 2, 3, 4, 6, 8, 12, and 24 as factors. Thus, Bajor has 24 Bajoran hours in a Bajoran day (the time it take Bajor to rotate on its axis once), and one could use military time just as they do on Earth by subtracting (or adding) 12 to convert from one to the other. But as it turns out, 24 Bajoran hours is surprisingly close to 26 Earth hours, whose length may be a Federation standard. Obviously, the locals use both Earth/Federation and Bajoran systems, and military and non-military times, and one knows which by context. Each Bajoran hour is therefore 1.083333 etc. Earth hours long, and there are 24 of those in the time it takes Bajor to rotate once on its axis.
I doubt the universal translators (which are apparently implanted inside their head (they were for the Ferengi, anyway) are an issue, mucking up conversations with extraneously precise conversions between languages, give me a minute translates to give me 1.04567 debars, or whatever Bajorans use in their native language. Numbers are more universal and objective and less subjective than words, and an agreed upon time system is likely in play - though where they are playing at any given moment may determine which system they are using.
I'm pretty confident since 26 is close to 24, humans and other Federation personnel make the adjustments to a 26-hour day cycle to keep in synch with the local population and local time. If the local day is wildly different from 24 standard Earth hours, that may be much harder to do, but an extra 2 hours a day is probably not that hard an adjustment to make - though like jet lag, it can take time to make the adjustment.