I'm not sure I understand your question. Your data is just data and is not associated with a particular OS. Your old computer and your TM backup drive should not be affected by the migration. The migration will not change the OS on the new computer. Do not choose the "Restore" option with the TM Lion backup on the new computer.
When boot your new computer for the first time Setup Assistant will ask if you want to migrate from another computer or from a TM backup. You do not want to choose to migrate your applications because they are much too old to be compatible with Catalina.
Here is an Apple article about setting up a new Mac and transferring your documents using Setup Assistant:
https://support.apple.com/en-us/HT205749
Using Migration Assistant if you've already finished setting up your new Mac:
https://support.apple.com/en-us/HT204350
I don't use TM, but I think it would be best to use a new hard drive and start a fresh TM backup for the new system. You don't want to erase the original TM drive too soon in case you encounter issues. Plus if your TM drive is old and well used, then a fresh drive is a good idea since you don't want to risk a drive failure when you really need the backup.