I've been hearing about Android 5.1 for awhile now. However, I've been having my Nexus 6 since January on Verizon's network (cut sim card, etc.), but I have yet to receive a 5.1 OTA update. Am I missing something?
Updates are staggered - it's a few months at least between the first one sent out (which is long after you hear about it) and the last one has been sent out.
I forced the update on my AT&T branded phone by removing the sim card and doing a factory reset. When the phone restarted it prompted me to download the latest update. Hopefully, it will work on the Verizon version. MAKE SURE YOU BACK UP YOUR DATA FIRST!!
see link below for article with instructions.