I too encountered the exact same problem after updating to 19.10. Here is how I (just now) resolved it:
First, you have TWO problems, not one. Both your installation is screwed up AND the Grub Bootloader is messed up. And running only one fix won't fix everything. You need BOTH the "boot-repair-disk" and the latest version of Ubu (both on USB boot drives. DON'T use a DVD.)
If you try to do (only) a "Repair Install" from the Ubu Live disk first, you'll still be greeted by the "grub rescue>" prompt when done. :( So FIRST you must use the "boot-repair-disk". Tell it to repair your broken boot partition with Ubu on it. If you aren't sure of the partition ID, launch "GParted" from the "Start" menu (bottom left.)
Repair that boot partition. This should at least bring Grub back. Try to launch Ubu. If it works, you're done. If not, boot the Live "CD" from USB.
Double-click the "Install Ubuntu 19.10" icon on the desktop (don't worry, there will be an option to repair w/o losing your old programs/files.)
I recommend checking the boxes to download all updates during install, including 3rd party.
The Installer should detect your broken partition and give you the option to repair it (the first option.) It may need to disable some 3rd Party repositories. Not a big deal, they're easy enough to get back later.
(Note: If you had to login with a password before, don't try to select "login w/o password" now. It won't let you in when you're done.)
Once done, you should have Ubuntu 19.10 installed with all/most of your existing apps still installed (though the toolbar shortcuts will be reset.) I had to reinstall a few 3rd party apps, but their configurations were still there afterwards, so nothing was lost. :)