During the spring and early summer, most small gas engine / mower repair shops are swamped with work. 2-3 weeks wait on a repair is about par for the course (unfortunately). This is true of all types of small gas engine shops, not necessarily any brand.
When Honda replaces a part on a mower under warranty, that part, along with all the other parts on the mower, is covered under the original factory warranty. So, if you had two years warranty left on the mower, the replacement transmission (
when installed by the dealer) would also have two years of warranty.
FYI, if you were to go to a Honda dealer and buy a new transmission, such cash & carry parts have a six-month warranty.
Full warranty details here at Honda's website:
Honda Power Equipment Warranty