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.
Pleased you got it sorted, thankfully my Nexus 6 was fine after the Android 6 update. I hope your finding it a nice experience. I'm getting loads more battery life, probably an extra half a day with my usage. The new optimisation is excellent. Shame they kept the white theme, I was so looking forward to an option.
What's your view on the new Nexus phones? I usually change mine every year, having previously a 4, 5 and 6. Not sure if there is enough of an upgrade to the 6P for me TBH.