Interesting solution to my issue: I thought I would swap the old unit back and see if it would work, and no signal there either! The only thing that had changed was I had used a new micro-HDMI cable with the new unit, and just kept it when I swapped the old one back. The old had been using a mini-DisplayPort cable. Hmmm. Switched to the mini-DP cable, put the new NUC in, and all is fine.
I don't think the HDMI cable is bad - it works fine when connected directly to a monitor - but I do think there is maybe a priority amongst the video ports, maybe? Anyway, problem solved for me.