how to reinstall Catalina from big sur with no time machine

Gulp, I'd try tis...

Start up from macOS Recovery

To start up from macOS Recovery, turn on your Mac and immediately press and hold one of the following combinations on your keyboard. Command-R is generally recommended, especially if you never installed macOS Sierra 10.12.4 or later… https://support.apple.com/en-us/HT204904#notupdated

Command (⌘)-R

Install the latest macOS that was installed on your Mac.

1.

Option-⌘-R

Upgrade to the latest macOS compatible with your Mac.

Shift-Option-⌘-R

Install the macOS that came with your Mac, or the closest version still available.

2. Decide whether to erase (format) your disk

If you need to erase your disk before installing macOS, select Disk Utility from the Utilities window, then click Continue. You probably don't need to erase, unless you're selling or giving away your Mac or have an issue that requires you to erase. Learn more about when and how to erase.

3. Install macOS

After starting up from macOS Recovery, follow these steps to install macOS:

1. Choose Reinstall macOS (or Reinstall OS X) from the Utilities window.
2. Click Continue, then follow the onscreen instructions to choose your disk and begin installation.
3. If the installer asks to unlock your disk, enter the password you use to log in to your Mac. If it doesn't see your disk, or it says that it can't install on your computer or volume, you might need to erase your disk. 

1. Please allow installation to complete without putting your Mac to sleep or closing its lid. During installation, your Mac might restart and show a progress bar several times, and the screen might be empty for minutes at a time.

If your Mac restarts to a setup assistant, but you're selling or giving it away, press Command-Q to quit the assistant without completing setup. Then click Shut Down. When the new owner starts up the Mac, they can use their own information to complete setup.

If you never installed macOS Sierra 10.12.4

If macOS Sierra 10.12.4 or later was never installed on your Mac, macOS Recovery works differently:

• Command-R is still the recommended way to start up from macOS Recovery. This combination makes sure that the installation isn't associated with your Apple ID, which is important if you're selling or giving away your Mac.
• Option-Command-R installs the macOS that came with your Mac, or the closest version still available.
• Shift-Option-Command-R isn't available.

https://support.apple.com/en-us/HT204904 

USB Installation & Wipe and Install

To perform this action will require booting from a Bootable Installer. 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 Big Sur would have the done on a computer that Qualifies to run Big Sur.

Once that is done read on for preparing the Destination computer  >> Special Notation - works on Intel Based Apple Computers & May on Apple Silicon M1 CPU

Extra Special Notation regarding the Touch ID equipped Apple Computer. About Startup Security Utility and Must Enable from Recovery Mode the ability to boot from External Drive Before Attempting 

The next step will Wipe ALL Data and there are no do-over - period.

Shutdown computer and disconnect all external drive Except the newly created Bootable Installer. Restart and immediately hold the OPTION key until the Startup Manager appears and choose the USB Drive. It will present options >> Disk Utilities >> View >> View ALL attached Drives. Choose the Upper Most Drive ( not the volumes indented and list below ). The drive normally is called Apple Media or Apple SSD - that is the drive to Erase and format as APFS with the GUID Partition Map. Once that is done >> backup out of Disk Utilities and choose install macOS. Follow the prompts and it may automatically reboot several time. Upon a final reboot - Setup Assist will present with the newer version of macOS.

Great to hear my friend, take care! :)