The seasons exist in Dragon Heist to allow for some replayability - each one comes with a different "main bad guy". It also controls how the encounters in Chapter 4 are stitched together to form the "main quest".
Chapter 5: Spring Madness doesn't really have any hard ties to the season. The main antagonist can be (and is) a lunatic at any time of the year.
Chapter 6: Hell of a Summer has some specific time cues related to a holiday and birthdays, but sufficient knowledge of the Realm's Calendar of Harptos could easily shuffle that around by picking a different holiday.
Chapter 7: Maestro's Fall likewise is tied to a particular event. This one is a bit more difficult to find another event to tie it to. It would be better to move the event to a different season, but note that a parade in winter is not terribly likely.
Chapter 8: Winter Wizardry has a winter-specific scene described in it, but it really doesn't have anything to do with the plot or the "main bad guy". It could easily remain in winter even if the "main bad guy" is different.