Adopting a time machine backup from an old mac instead of starting afresh

"I could just wipe the disk and start again, but I'd prefer not to in case there is something on it I subsequently need."

In a situation like this why don't you simply buy another external drive? They're cheap enough after all. It's also good to augment your disaster recovery plan with extra drives whenever possible. Drives can and do fail and its good practice to keep more than one backup if your budget allows. I always advise 3 as a minimum - grandfather, father, son. The more the better TBH.

". . . Is it possible for a "new" installation of Catalina to adopt a TM backup from a previous installation?"

TM is not something I use - too unreliable for me - but thinking about it, it should do? After all you would use a TM backup if you needed to reinstall the OS anyway, regardless of whether you format the drive or not. Perhaps others who are more familiar with TM will post and offer more advice regarding it and your question?

Good luck

I have also seen this happen recently. I did what Rocco suggested and got another hard drive to use and save the old one. I also found out that even if it allows you to use you cannot scroll directly back in time. Then you need to use the shift-command-c keys to move back in time. Just like mail no longer works with TM to scroll back starting with Catalina. Apple keeps tweeking the TM app to act differently over time for some reason.