If you were facing a WiFi issue downloading Big Sur what makes you think it will be any better with Catalina.
Whenever downloading from Apples Servers it is best to use cabled connection to you router.
Do not use the mac while it is downloading and do no use any devices that share the same internet connection.
If you can only use WiFi then move your mac as close to the router as possible.
Download Catalina from this link, remember to use your Safari browser.
How to get old versions of macOS – Apple Support
Go to Check Compatibility to see if your mac qualifies for the OS upgrade you want.
Then go to the Download OS section and click on macOS Catalina.
Before upgrading make a full back up of your Mac.
Check that third party apps you use have updates available for the new OS.
https://roaringapps.com
Check that your printers/ scanners etc have updated drivers and software available for the new OS.
Connect your mac to your router via cable rather than rely on WiFi.
Do not use your mac while it is downloading
and do not use any other devices that share the same internet connection.
If the App Store says ‘Open’ then it is detecting a previous download of the install app
go to the Applications folder locate Install macOS Catalina.app.
Move it to the Trash, empty the Trash.
Restart, click on the link again, the App Store should say ‘Get' or 'Download'.
If you are considering upgrading to Catalina or Big Sur be warned they do not support 32bit apps.
Your Mac is probably running 32bit apps and these will need to be updated to 64bit apps,
some updates may be free, some may be paid. Use this app to discover which apps you have
are 32bit, https://www.stclairsoft.com/Go64/ then contact the developers of the software to see
if they are providing 64bit updates or alternatives.