How can I go back to Yosemite after updating to Catalina?

macOS 10.13+ uses a new drive layout and file system which older versions of macOS do not recognize so you will need to erase the whole physical drive and reinstall macOS 10.10 before restoring from a backup.

You will need to use a macOS 10.10 USB installer. Here is an Apple article with instructions for downloading & extracting the macOS 10.10 installer:

How to get old versions of macOS - Apple Support

Here is an Apple article with instructions for creating a bootable macOS USB installer. However, the instructions do not mention anything about macOS 10.10 Yosemite, so you will need to follow the instructions for creating a macOS 10.11 El Capitan bootable USB installer and slightly modifying them to reflect "Yosemite" instead of "El Capitan":

How to create a bootable installer for macOS - Apple Support

Here is the command needed for the Terminal app if you are following the instructions in the Apple article:

sudo  /Applications/Install\ OS\ X\ Yosemite.app/Contents/Resources/createinstallmedia  --volume  /Volumes/MyVolume  --applicationpath  /Applications/Install\ OS\ X\ Yosemite.app

When booted from the macOS 10.10 USB installer you will need to partition & format the whole physical drive using the instructions in this article:

https://www.owcdigital.com/assets/support/support-formatting-and-migration/Mac_Formatting_6-10.pdf

After which you will need to install macOS 10.10. Then on first boot of the clean install of macOS 10.10, you can select to Restore from a Time Machine Backup.

Another option may be to install macOS 10.10 to an external drive (USB3 SSD would be best for performance) to dual boot if you can do the majority of your work from Catalina. However, macOS 10.10 will not be able to see any data on the internal Catalina drive since Catalina uses a new drive layout and file system which macOS 10.10 does not understand. I never recommend dual booting from the main internal drive as people usually regret it later since people usually choose the wrong partition sizes and realize neither OS has enough storage space.

Depending on the performance needed and the resources of your Mac, you may even be able to install macOS 10.10 using a Virtual Machine such as Parallels, but this option will share resources (CPU, memory, storage) with both operating systems (sometimes this is fine, but depends on your setup and workloads for each OS).

Would follow advise fro @HWTech first

Only as a Fall-Back position.

Scorched Earth Method

This method will WIPE ALL Data and there is No Recovery - Period.

To perform this action will require booting from a Bootable Installer. 

The Bootable Installer can Only be performed on an Apple Computer 

This will have to be performed from a Qualifying Computer to run the version of macOS to be made on the Bootable Installer.

Example : Bootable Installer of Yosemite would have to be done on a computer that Qualifies to run Yosemite.

Notation: If the computer being used to perform this action is Too New or Too Old to qualify to run the version of macOS - this computer can not be used.

Alternative is, to gain access to a Qualifying Apple Computer from a family member, friend or associate.

Once that is done read on for preparing the Destination computer  >> Only works on Intel Based Apple Computers.

1 - Shutdown computer and disconnect all external drive Except the newly created Bootable Installer.

2- Restart and immediately hold the OPTION key until the Startup Manager appears and choose the USB Drive. 

3 - It will present options >> Disk Utilities >> View >> View ALL attached Drives. 

4 - Choose the Upper Most Drive ( not the volumes indented and list below ).

5 - Formatting for macOS 10.13 High Sierra and below requires HFS Journaled with the GUID Partition Map

6 - Once that is done >> backup out of Disk Utilities and choose install macOS. 

7 - Follow the prompts and it may automatically reboot several time. 

8 - Upon a final reboot - Setup Assist will present with the newer version of macOS.