I did the same thing, 3 weeks ago. Series 4 on AT&T LTE. I cancelled the AT&T on watch a couple weeks before we switched my whole household full of phones and iPads over to Verizon. HOWEVER, AT&T does not do prorated cancellations now, so they explained my watch would be active through the 14th, the last day of my billing cycle. I waited a few days beyond that to activate on Verizon. Went easy enough just using my phone.
I think some folks have heard you can't switch providers on the watch, particularly if you've activated on AT&T. It's not true, but I wonder if you have to wait until AT&T cancels it in their system, at end of that billing cycle.
Incidentally, I'm in North Fort Worth, in the middle of the DFW metroplex, 50 feet off Interstate 35, and AT&T service sucked. Verizon service, including on the watch, is at or near full signal everywhere, and data speeds are incredible.