Failed update to 18.04, no GNOME or any desktop

Solution 1:

The following list of actions solved the problem:

sudo apt-get update
sudo apt-get install ubuntu-desktop
sudo apt-get install gnome-core lightdm
sudo dpkg-reconfigure lightdm # select LightDM

then OP successfully logged in to the session.

Solution 2:

It is very probably related to an online update that was rolled out yesterday by Ubuntu.

People who have been upgrading to 18.04 yesterday are experiencing it (I cannot boot normally after online update 23.8.2018 from 16.04 to 18.04)

People (like who) have been using 18.04 but applied the patch are experiencing it (Debian login screen after updating Ubuntu 18.04)

I had the same problem as you on the second screenshot. The GDM starts but somehow cannot finish loading. I also noticed errors with amdpro and radeon module loading in the kernel log.