My sleep got better within two days after starting progesterone (200 mg nightly, which is more than many others), and then even better all the time with estrogen as I increased it. I probably reached the best sleep on my original regime after about 6 months because I could only increase estrogen a little bit at a time (my clinician wanted me to wait a month or two each time to increase my dose).
Looking back, I would have asked to raise my dose of estrogen every month if sleep wasn't good. My clinician encouraged me to keep increasing estrogen until I got back to sleep comparable to before perimenopause....I don't know if I am fully there, but it went from hellish to okay in a few months, pretty good in 6. I am on a higher dose of estrogen, though, so it may take less time to get up to the ideal dose for others. My sleep also wasn't good on the 100 mg of progesterone many take; I had to take 200 mg nightly -- with food -- to sleep well.
Kara