I have a theory that the ‘Scream of the Shalka’ Doctor is in fact a previous Doctor working for the Division (ie. before the Fugitive Doctor), and another about the Master in it as well, but I’ll get to that in a bit. The ‘Shalka’ Doctor says to himself as he arrives that “they” (almost definitely the Time Lords) had sent him there, and has a feel of reluctance about it; definitely doing his job, but not entirely comfortable being ordered around all the time - very much like the Doctor. There’s nothing mentioned in the story that directly contradicts this theory, which is more that can be said for a lot of established aspects of the show.
I think that the TARDIS being a Police Box in this as well kind of works in a way since the Fugitive Doctor also had a Police Box TARDIS - something which I think was very much overlooked on Chibnall’s behalf. This, however, kind of gives it a bit more credibility, in that the TARDIS was stuck as a Police Box for the Division Doctors (probably issued to them brand new), but one day got stuck as a Police Box, but when - centuries later - the First Doctor stole it, it reverted back to that as a kind of memory or continuation. This goes back to the idea that the TARDIS is sentient and knows what’s best for the Doctor, as shown in ‘The Doctor’s Wife’.
Now for the Master…
The ‘Android’ Master in ‘Scream of the Shalka’ said that he was given the choice to die or become, essentially, the Doctor’s pet. What if the Division contacted the Master just at the moment of death (as the Time Lords did Clara), and offered him this choice. Then suppose that this moment was in fact when Missy ‘dies’ on the Mondasian ship. This would mean that the ‘redeemed’ Missy was offered to help a previous version of the Doctor or face death. Knowing the Master, the non-death option would always be taken, but it seems almost perfect that Missy is given this chance to do what she was working for over millennia to do. This also explains that when we see the Master next in ‘Spyfall’, he’s back to his mad old self - the Master was ‘recruited’ and therefore learned about the Division and the Timeless Child and so on, explaining how he came across the information in the first place (I know it says he hacked the Matrix, but how or why he did that, and especially at this point in time, is beyond me). I feel like this ties in nicely with the already established Timeless Child timeline, but also gives Missy’s ‘redemption arc’ the send off it deserves. It’s also just nice to be able to fit ‘Scream of the Shalka’ into the Doctor Who timeline because it’s just great.
If you’ve made it this far, I have to say thank you and also well done. Let me know what you make of this if it hasn’t just been paragraphs of sheer nonsense.