There are plenty of how to instructions on the Internet for replacing the battery in your NUC.
None of them talk about issues with batteries on the Internet.
If you have a battery that doesn't work, chances are it is just wired up back to front.
For example this battery is from EBAY.
However take a close look at the plug vs your old batteries plug.
Luckily this is very easily fixed. The metal pins that are crimped on the ends of the wires are held in place in the plug with the two plastic tabs that you can see in the image above.
Simply lift the tag gently (do not bend or break the tag) and then while it is being held up pull on the wire to release the pin.
Now the pins are out, carefully insert them back into the plug in the correct order.
They should click back into place.
