OK, so I took my Model S "permanently" off Wi-Fi today - this was the second time a major delay in getting the update was ended by switching to 3G.
It may be coincidental, or it might something in my private Wi-Fi setup (great bars though), but I'm going to give this a try, because for all appearances my 6.2 came quite late by TMC standards - as did 6.1.
Both had major delays and both came once I was in 3G territory for a sufficient time, not after being on Wi-Fi. I had waited for 6.2 for weeks and within 12 hours of turning off Wi-Fi the update was available.
I see no benefit in using Wi-Fi either in my use, so there seems to be no harm in trying if this helps at all.
Just my personal experience, and may be coincidental of course.