Paramount+ purchased via Apple not playing on new TV

Define "via Apple".

What exactly did you do?

Did you subscribe to the Channel in Apple's Tv App?

Or did you subscribe to the Paramount+ standalone app, but are paying the subscription through your Apple account?

If you subscribed through the channel in the TV App, you need to link that subscription to the Paramount+ stand alone app.

click here ➜ https://support.paramountplus.com/s/article/PI-How-do-I-sign-into-Paramount-with-my-Apple-TV-Channel-subscription?language=en_US

If you signed up through the Paramount+ app, you will need to log in to the stand alone app on the Tv with your Apple account.