Upgrade to Catalina erased all data, have backups but how to restore applications/software?
The Migration Assistant is the program on any Mac that allows you to transfer in or out data and applications. I would connect your backup drive and then start the assistant.
It will then show you the information from the most recent backup or if there are many backups and the system can’t tell which backup was on your Mac, you can choose from the many destinations that can be saved to one Time Machine drive.
You can bring in accounts, settings, applications or a mix of some or all of the above.
Here is Apple’s support article that I think does a good job of the overall design and intent of this process.

https://support.apple.com/en-us/HT204350

Since Catalina has excellent support for containers, the old steps to reinvoke the assistant are a bit complicated. If you don’t have anything on your new system, erasing the content / reinstalling in an erase install might be much easier now that you are on Catalina.