Planning
Most of the wait is before anyone arrives
Replacement windows are built to the measurements of your openings. The installation is the short part.
Where the weeks go
Measuring happens once the order is agreed, and it is done by the people who will fit them rather than by whoever quoted. Manufacture then takes weeks rather than days, and longer for anything not in a standard size, for special colours, and for impact-rated units.
Installation itself is fast: a crew commonly fits several openings in a day, and a whole house of straightforward replacements is a matter of days, not weeks. Full-frame work is slower because the trim has to be rebuilt.
The practical consequence is seasonal. Ordering in late winter means fitting in spring; ordering when the first cold arrives means waiting through part of the winter you were trying to fix.
Permits
Whether a permit is needed depends on the jurisdiction and on the work. Like-for-like replacement in the same opening is exempt in some places and not in others. Changing the size of an opening, altering structure above it, or fitting an egress window in a bedroom or basement almost always requires one.
Coastal jurisdictions are stricter, and there the permit is also where the product approval is checked.
The permit is normally pulled by the contractor. It is worth asking who is pulling it and whether its cost is inside the quote — a permit taken out in the homeowner's name puts the compliance on the homeowner.
What to ask before signing
The lead time in weeks, in writing. What happens if a unit arrives damaged or mismeasured. Who pulls the permit. And whether the price holds if the measurement finds rot behind the trim — that is the most common reason a quote changes.