Right-click:
If you have multiple display outputs, they'll all appear there, and you can choose which one goes to the nursery TV.
Keep in mind though, that it's silent. You'll have to send the audio there separately, via OBS's Monitor. If you're already using the Monitor for your headphones, you'll have to choose one or the other. Or you might try this plugin for multiple physical outputs:
Plugin for OBS Studio to add Audio Monitor dock and filter. It allows you to put the audio of a OBS source to an audio device by adding the Audio Monitor filter on the Source in OBS Studio. This can be useful in different use cases...
obsproject.com
Either way, you send a silent picture to that physical HDMI connector as above, and then you send the audio separately to that same HDMI connector. Or VGA + line out, or whatever you've got.