The thing to remember with all the UNIT fuckups here is that this wasn't UNIT's actual history- this is the Grand Serpent right now altering their history to give the Sontarans a victory. So the fact that it's all wrong is, well, the entire point of that.
Except the Serpent firing Kate seemed to be the backstory to UNIT being out-of-service during "Resolution" a few years ago, but I guess when you're talking about changing history, you can draw the line anywhere you like between "unaltered" and "altered" in the absence of an explicit statement. Though how UNIT being shut down as a weird Brexit joke in 2017 squares with the Serpent still being in-charge in 2021, and UNIT still having control of the world's entire nuclear arsenal, I can't tell you.