I'd be willing to bet that if they do a UNIT spinoff, it will be a special detachment with a specific mission, and Donna will only appear in cameos. Kate Stewart might be like an Amanda Waller, monitoring their progress from HQ and only rarely getting directly involved.
With the amount of focus it got in The Giggle, the HQ is absolutely playing a big part in the spinoff. They made a whole robot character who appears to be anchored into the HQ, and since the robot served no purpose whatsoever in the plot of The Giggle itself, the only reason to make the robot is because it will be a regular part of the spinoff. Therefore, the spinoff is going to be centered on the HQ we saw.
It is going to be interesting seeing how they balance it (and actually in the main Doctor Who show as well) when there's literally an entire second Doctor just sitting there doing nothing. Like anytime they escalate things to world-threatening stakes, it's going to feel weird that there's a version of the Doctor there who could absolutely help but just... doesn't for some reason. Like I know this Doctor's supposed to be recovering and taking it easy, but it seems a bit out of character for any version of the Doctor to not help when people are in trouble.
I would think they'll play it that he is off somewhere on an adventure of vacationing somewhere.
A good old fashioned Doctor Who Community