Processes in Microsoft 365 for setting up Office apps, redeeming product keys, and activating licenses.
Go to where the xml files are stored.
Right click one and open with Notepad.
In the top lines will be the url or UNC which it uses to find the template.
That will tell you where to look.
Go to that location, it could even be cached, but hopefully it is there. Right click on it and select design. Then do a save as to your my documents or something. Now you can transfer it to the new laptop and republish the the same location it was before.
Feel free to post what you find.