The Doctor dismisses the activation of the HADS as it being accidentally reenabled as part of the TARDIS rebuilding. But an alternate explanation is the TARDIS realized its presence was the worst possible thing for the situation, because, as an intelligent being, the Not-Things could have copied it and gone anywhere and anywhen — that is the danger they pose.
First and foremost, if the TARDIS knows that its presence is dangerous, since they can copy its thoughts, then it would also realise that translating the message is dangerous. After all, if the Doctor knows it's counting down, he's going to think about why it's counting down, and realise it's a bomb timer a lot faster than if he needs to get to the cockpit. Turning off translation gives Jimbo time to get in place without the Not-Things stopping it.
And second, the TARDIS is exploding, and in the process of repairing itself. That's (theoretically) what reactivated the HADS in the first place. So, it stands to reason that the console damage may have forced it to turn off the one or more components of the whole translation system, and the Translator Microbes probably aren't all that useful unless everything is working properly. It's not the TARDIS leaving that stops translation, it's the TARDIS self-repairing that stops it.
It makes sense the Not-Things struggle with proper human proportions: the number of body parts they have, the fact that stuff keeps existing when you're not looking at it, etc. It's all stuff people know, but almost never think about, thus it takes longer for the Not-Things to copy the knowledge.
Assuming that Donna is telling the truth about not remembering specific details about the Doctor’s life, it would still make sense for the Not-Donna to know about the Timeless Child — who’s to say the two beings didn’t swap forms based on what was convenient at the time?
More fridge tearjerker, but given that the Tenth Doctor defined himself by being "the Last of the Time Lords", it's no wonder the revelation of him being the Timeless Child struck a nerve with Fourteen.
The Not-Things explain they're so violent because somehow the "blood, fury, and hate" of the universe was so great, evil has infected the very edge of creation itself. When told life is more than that, Not-Donna says, "Love letters don't travel very far," meaning even with the Doctor traveling around, helping wherever they can, the universe is still more evil than good. We already knew the Whoniverse is objectively a horrible place, but this makes the Doctor's struggles seem borderline futile.
Unless, of course, you consider that at the very least, the Doctor is making some difference to somebody, which is better than nothing. But even with them copying his intelligence, the idea is sadly still lost on the Not-Things.
Besides, there's also the question of exactly where and when the Not-Things are in terms of their temporal and spatial location; it may be that they're from a time and place where the Doctor just hasn't had much influence, so things are better in other parts of the universe and they just found a particularly bad part of it.
Who's to say the Not-Things were actually destroyed by the explosion at the end? They're able to drastically distort their bodies and make objects randomly cease to exist, either of which could have enabled them to survive an explosion and the vacuum of space. They'd have a long way to travel to go anywhere, but they now know what bombs are.
While the Not-Donna nearly escaping on the TARDIS seems like the lesser of two evils compared to the Not-Doctor seizing it (other than leaving the real Donna to die, of course,) this actually would have been worse. The Not-Donna shows that she has all the memories of being the DoctorDonna, a being that is as brilliant as and possibly exceeds the capabilities of the Doctor himself. An evil DoctorDonna in charge of the TARDIS would have been disastrous for the universe.
