Don't just look at the alphabetical listings -- examine the entire directory for clues.
The best internal evidence I have for the lead time so far is in some of the Price and Lee directories from New England which have lists of deaths and marriages from the previous year. Those run from December to December. E.g. 1890 Holyoke Mass has Deaths from Dec 1, 1888 to Dec 1, 1889.
Another thing to consider is whether any given directory is consistent with itself. I have a couple of cases where people have 'disappeared' from the alphabetical listings, but their entries in the Street Index (the numerical section) linger. Had they moved, and the street listings not caught up with the alphabetical listings, or did they not pay for a listing in the alphabetical section? It's difficult to know for sure, and like any source, we are left with the task of analyzing the information in the directory.
Check the front matter of the directories to see if the directory has any information about how the information was collected. Do research to determine whether the locality in question had a standard "Moving Day" where new leases started. The directory publishers would have needed to start canvassing residents after moving day (otherwise the information in the directory would have been too stale) but they couldn't take too long about it, or the businesses relying on the directories would complain, as we can see from this snippet taken from the trade publication Directory Bulletin, Volumes 1-5. The screenshot below is from a section "Notes From the Field":
Considerable dissatisfaction has been expressed with the length of
time taken to issue the directory heretofore. Moving day in
Birmingham [Alabama] is September first; and the people want their
directory delivered within four months from that date.
Other entries in "Notes from the Field" list the expected pub dates for directories in various states.
Historical newspapers might also run articles or notices advertising the release date of the directory, or have news if a particular year's issue was delayed.
I am indebted to Dan Earl and Mirriam Robbins for their helpful comments on my question posted in the Historical City Directories Facebook group. The link to the Directory Bulletin is courtesy Mirriam Robbins, webmaster of the Online Historical Directories Website and Admin of the Facebook Group Historical City Directories.