I am having iMessage and connectivity problems with iOS 8.1 (presumably not the fact that it is an iPhone 6 plus phone) as well. I tested WIFI vs LTE and both were non-functional. iMessage would not go through, but worked when switched to text (also received a phone call during my troubleshooting). Verified WIFI functionality from a laptop (and good speed). Switched out of WIFI to LTE (Verizon network) and I still could not access.
After RESTARTING the phone, everything magically worked. Based on the other posts I've found Googling, this appears to be a "known" issue as multiple folks are reporting it. There were problems with all iterations of iOS 8.x so far (there's been 4 or 5 versions, I've lost track).
Additionally, I've had a lockup that required me to restart the phone (might have been connectivity related (or not). This manifested itself as a "dead zone" at the top left of the screen (I could otherwise still navigate via other portions of the screen, but there's quite a bit at the top left, so I restarted. Net result the same, everything magically functioned after the restart.
In short, this is an issue with iOS 8.x, conclusively. Not the provider, not your router, likely not the phone itself (haven't had the 5 or either of two 5c phones flake out on me, so I don't explicitly have test data from/for them). There are multiple threads here reporting this issue.
There is this: http://www.iphonehacks.com/2014/10/problems-ios-8-1-update.html and there are older threads for previous iterations of iOS 8.x, such as this: http://www.techtimes.com/articles/16718/20140930/apple-ios-8-0-2-still-problemat ic-users-report-connectivity-issues-other-glitches.htm While some of this might be helpful, such as this one: http://osxdaily.com/2014/09/22/ios-8-wi-fi-problems-fix/, the issue ultimately is the OS itself (settings changes are irrelevant if restarting the phone fixes the issue). Compounding the problem is the inconsistency in troubleshooting/reporting as the result of so many versions of this OS being pushed out at a rapid pace.
APPLE: At a minimum, a response in some way, shape or form would be appropriate. "Hey, our bad, we're working on it" or similar warm, fuzzy statement. There's enough out there where we can be clear it isn't "just us". Thanks in advance!