(Note: Addressing the general case of power-loss during an upgrade, before it reaches a state telling you that the upgrade was successful. Applicable to all releases.)
You said it yourself:
That tells me that
- the upgrade process did not complete,
- which in turn did not give you a complete OS installation or configuration,
- which means that a "sane" installation did not happen,
- which is what prevents LightDM from performing its portion of the startup process successfully.
If you don't have a second bootable drive, you would need to boot from a "Live" ubuntu ISO on either a CD or USB stick.
Open a MATE terminal session, then, from there, you need to do a proper full backup of all contents of all partitions that you have/had on the disk drive where the installation failed and copy it onto another disk drive ... so that you can reclaim and restore the good parts from that backup after you repaired the OS.
Once that data is "safely" put aside, because of the "corrupted" installation, you need to make a "fresh installation from scratch", not an upgrade.
Then, again from the terminal session, you need to collect information about your old configuration (if that is possible) to be able to rebuild the same disk configuration (IF that is what you want) to be able to answer questions during the disk setup step (during which it asks you to set up partitions on the disk), before attempting to do the fresh "install".
Only when that is all done, and you feel you have enough to recreate what you had before, THEN, and only then, you can start the installation process for a clean NEW installation.
If that is done right, all non-root partitions which you previously had should still be there, with their old content, as long as you did everything correctly during the "disk setup" phase and chose "different layout", which will show you what is on the disk as it is, and you can tweak what is there to fit your needs.
If the installation completes to the point of asking you to reboot, only then can you say that you have a bootable OS ... and the UbuntuMATE installation that will reboot properly, at which point it should never behave in such a way that you would become aware that LightDM was even involved in the process. 