OP wrote:
"JUst spoke to Apple support and they say that there's no way to reinstall Mojave and that the only way forward is to install Catalina."
That is wrong.
You
CAN "go back".
But because you did not create a bootable cloned backup before you upgraded, this is going to be
a long and winding road.
It sure helps when you tell us WHICH MAC you have and WHAT YEAR it was made.
Are you prepared to do all the work?
First, if your current TM backup is not working (this is why I will NEVER recommend that a Mac user use time machine, it can fail in "moments of need"), then the first thing you need to do is get ANOTHER external drive to create a new backup (yes, even if the internal drive is "munged up" a bit".
DO NOT USE TIME MACHINE to create this drive.
Instead, use CarbonCopyCloner, which is free to download and use for 30 days.
Get it here:
www.bombich.com
It's easy... just launch CCC and create the cloned backup.
Then, put it aside for now.
Now, you have to get ahold of a copy of Mojave.
You can try to get it from Apple here:
another link to try:
Both of the above are from my archives, they are not guaranteed to work.
You could also try:
Download and install current or previous versions of the Mac operating system on compatible Mac computers.
support.apple.com
If NONE of the above work for you, there are other, non-Apple-authorized sources from which you can get an installer -- but
I can't say any more about that here.
Once you HAVE an installer, I believe you're going to have to create a bootable USB flashdrive in order to boot the Mac, because I don't believe you can get the Mojave installer to launch while booted from Catalina.
To create a bootable USB flashdrive, I'd suggest you use a flashdrive 16gb or larger.
Easiest way to make it is by using a free app called "DiskMaker X".
BE SURE YOU DOWNLOAD THE RIGHT VERSION.
Once you get the flashdrive made, boot from it by rebooting and holding down the option key until the startup manager appears -- then select the flashdrive with the pointer and hit return.
Once you get booted, go to disk utility.
Choose "show all devices" from the view menu (VERY IMPORTANT step).
Now, look on the left.
The "topmost" item you see represents the physical drive inside.
Click on it and choose erase.
Choose APFS with GUID partition format.
Once the drive is erased, close disk utility and open the OS installer.
Begin "clicking through".
The install may involve one or more reboots, the screen may go dark for a minute or two, and you may see one or more "progress bars".
IMPORTANT:
IF the installer fails (may say it doesn't have the resources or something like that), you may need to "set back" the date.
Open terminal and enter:
"date 010112002019" (leave out quotation marks)
Then try the installer again.
Once the install is done, you'll see the initial setup screen (choose your language).
Begin clicking through.
When you get to the part where setup assistant asks if you wish to transfer data from another drive, NOW is the time to connect the tm backup and see if it works.
BE AWARE that it may not work at all.
But try it anyway.
IF IT DOESN'T WORK...
Then try connecting the CCC backup and see if setup assistant can grab anything from that.
IF THAT DOESN'T WORK...
Then finish setup anyway.
You will have to "rebuild your apps and data from scratch", using the files on the CCC backup.
MAKE SURE before you start copying over files that you do a "get info" on the CCC backup and (at the bottom of get info) put a check into the box "ignore ownership on this volume".
As I said, a long and winding road.
Next time, THINK before you upgrade on a whim, ok...?
