Ok, dumb question, but it could save time - have you physically compared the connectors of the old and new batteries?
I have a couple of these batteries in my spare parts pile - unfortunately none that I can positively say came from my failed BN unit. The keying is different from that of the replacement battery pictured above.
If you have wires in opposite positions, it is ~fairly~ easy (sarcastically says the guy who has done it many, many times) to swap the wires in the connector. Within that connector (positioned as you have it in the picture with the pen), there are two tiny white tabs that hold the pins into the connector. Using point of safety/sewing pin, you can lift the tabs slightly and the pin can be pulled from the connector (aside: to lower frustration level, I use tip of X-Acto knife to do this as it will lift and hold up the tab for the time it takes to get the pin pulled out). Once the pins are out, you can, one at a time, push them into opposite holes in connector. Pushing pin in and pulling back, you should feel when the tab locks it in place.
Once you have installed a battery with correct polarity, you should boot up using the F2 key to enter BIOS Setup. Within BIOS Setup, all you need to do is set the date and time and then you can exit. On the resulting POST, you should not see any complaints about the CMOS. To verify completely, after booting into Windows (or Linux), do a full shutdown and then unplug the NUC's power cord from the wall (A/C power) for 10 minutes. If all is good, upon powering up you should not see any CMOS complaints.
Hope this helps,
...S
