This message was posted on behalf of Intel Corporation
Thank you for joining the Intel® NUC Community. I am sorry to hear you are having issues with this matter.
If by mistake the Windows® 10 partition and the recovery partition from was deleted then unfortunately, there is no way to get the product key back. Microsoft® policy does not allow us to send a product key. You could get a Windows® 10 image and reinstall the operating system, or you can get in touch with Contact Support and check the replacement options for the unit.
If you haven't mucked up the image on your eMMC and can still boot to it, you can run a tool like ProduKey. It will display the keys for all of the Microsoft products installed. There are lots of (also freeware) tools out there that have similar capabilities.
Hope this helps,