The idea which I am hoping will still work is to use recovery and the Time Machine backup to go before the installation of Catalina. You can only do this from recovery. So choose a date and restore the entire backup to your External drive. Or even your internal drive if you so wish as long as you have tested the external drive with your current boot install actually boots properly.
You will need to erase the drive before you start the restore and then leave time machine to restore your previous working install. That means you don't need Mojave OS .. since it will be there in the backup. Once inside Mojave I think you have better chance of downloading it again.
If that does not work.. you certainly still need a restore of your previous OS as you cannot migrate easily from Catalina back to Mojave.
Apple might be able to help you if you ask nicely for a boot USB with Mojave.. There are other sources.. google around. Lots of Apple service centres will happily provide you with a Mojave boot USB for a few dollars if you have one near you.
Be assured plenty of people get themselves into the same boat so there are usually a good few articles on how to download the installer. Terminal commands should still work. The first one has a terminal command which only works in Catalina to download mojave.. give it a try.
eg
https://forums.macrumors.com/threads/where-to-download-macos-mojave.2205013/
https://osxdaily.com/2018/09/29/download-full-macos-mojave-installer/
Some will be out of date.. but don't give up.
Also note the instructions to create your own boot USB installer is important. Once you run the downloaded installer it is deleted. And copying it to another location often seems to break it. So my advice is to create the USB immediately. Put a label on it and tuck it in a draw so you always have it.