It's highly unlikely that iOS 9.1 has anything to do with this issue. If there was a wide-ranging cellular connectivity issue with 9.1, a lot more people would be complaining. Most likely something just went wrong in the upgrade process - a settings file got corrupted or something to that effect. reverting to the older version is not the solution you're looking for.
Doing the things C DM mentioned in his initial post, in the order he mentioned them (checking after each step) will almost certainly resolve the issue.
I wouldn't use the last backup, as the corrupt file may be located in that backup. Try resetting network settings. - Settings->General->Reset->Reset Network Settings. If that doesn't work, I would restore it and set it up as new. A pain but will most likely solve the problem.