Another update: I think I've ruled out nvidia-persistenced being started via udev as the possible culprit.
I tried Trusty with the nvidia-343 driver from the System76 PPA (which starts nvidia-persistenced with udev)... and I can connect to WiFi just fine.
I also tried an up-to-date Vivid install with the same nvidia-343 driver... and again, I can connect to WiFi just fine.
So something specific to Utopic is causing the problem (whether or not that problem is actually in NetworkManager itself, I still don't know).
Another update: I think I've ruled out nvidia-persistenced being started via udev as the possible culprit.
I tried Trusty with the nvidia-343 driver from the System76 PPA (which starts nvidia-persistenced with udev)... and I can connect to WiFi just fine.
I also tried an up-to-date Vivid install with the same nvidia-343 driver... and again, I can connect to WiFi just fine.
So something specific to Utopic is causing the problem (whether or not that problem is actually in NetworkManager itself, I still don't know).