Argh that sucks. And strangely, quite interesting... Things to try:
1). Have you tried connecting a pair of earphones / headphones to the front jack? Still a no-go?
2). You tried your T20 with something else I assume (to rule out the T20 is faulty; I doubt it, but to be sure)
3). Have you tried the T20's front AUX input as well? (again, to rule things out).
4). Audio settings in the OS. If you're using Windows, go to the Control Panel, then Sound.
4a). In the small-ish settings window that pops up, there should be two audio devices showing (one should be HDMI, the other, assuming the 3.5mm cable is plugged in, the front jack).
5). Right-click the device you want and make it the 'default', click OK, and see if that works (may require a reboot).
Yeah. I recall reading elsewhere that when one plugs in something in the front jack the NUC switches the audio output (from HDMI) to the jack automatically. The above just manually forces the change provided the audio jack at the front is populated. Hopefully it works out.
P.S.
What sort of setup are you going for with your NUC? As mentioned previously, I'm planning on running mine with an AOC display which has an audio pass-through. I've used this setup with a PS3 and a Vaio Z and my T40ii worked fine.