I recently updated my LG Nexus 5 to Marshmellow (Android 6). When I rebooted I lost the data connection.
I tried all the suggestions about removing and adding APN settings again, using all the variants suggested by the community, and rebooting the phone eash time. This had no effect, the data connection was not restored.
However, I've now got the data connection working again by putting the phone into Airplane mode for a little while. When I turned Airplane mode off again the data connection was restored.
I found the instruction on THIS PAGE.
Hope this may help anyone else having the same issue. 🙂