Julian states in 'Inquisition' that he was imprisoned for 37 days, correcting Sloan who states it was 5 weeks (so 5 weeks, 2 days total). The writers seem to stick with earth weeks the whole way through, which is plausible because it's a tv show.
In 'By Inferno's Light' Worf fights three times, the first time he breaks three, maybe four, ribs. The second time Martok announces that Worf won seven out of seven bouts, and the third time he faces the First and almost dies before escaping. It is *fairly* safe to assume these fighting sessions are on successive days, so the start of 'In Purgatory's Shadow' took place on day 34 of Julian's imprisonment.
Next we are going to jump to 'The Darkness and the Light', the episode where Kira is almost killed by a crazed Cardassian.
In that episode he goes to cut the baby out and Kira says: 'You can't bring it into the light! Not yet! Doctor Bashir said if I don't have at least three more weeks.'
This means that, as Kira went safely to term, 'The Begotten' must take place at least three weeks (21 days) after 'The Darkness and the Light'.
In 'In Purgatory's Shadow', Dax states that the O'Brien baby is 'less than a month old'. So that means 'The Begotten' must have taken place between one and four weeks previously. If we take it to be one week, then 34 - 28 = 6 days, making it most likely that Julian was taken just after 'Rapture', assuming that 'The Darkness and the Light' didn't start the day after Sisko's brain surgery. He seemed fully recovered and back at work, so it is likely that at least some time had passed between the two episodes.
If, however, we take the more likely scenario that baby Kirayoshi was closer to three weeks old (making Dax's rounding to 'less than a month' more plausible), then that would mean that 'The Darkness and the Light took place 42 days prior to 'In Purgatory's Shadow', meaning that Julian was still Julian when these events occurred.
He would, in fact, have been taken approximately 8 days after 'The Darkness and the Light' and 13 days before 'The Begotten'.
And finally, in 'If Wishes Were Horses' Julian canonically takes a nap in his uniform, pips, boots, and all. Best explanation for him being suited up despite being nabbed sleeping is that he just likes wearing his old uniform to bed.
Conclusion: Julian pulled on an old uniform to take a nap while at that burn conference sometime after making sure Kira was okay from the events of 'The Darkness and the Light' and his replacement stepped back onto the station just in time to be unable to save a baby Changeling for reasons that remain unclear, perhaps it was always impossible.
I hope it felt really bitter about that.
