Doctor Who: EXACTLY Where Jo Martin's Doctor Fits Into Doctor Who Canon
Now we get onto what I believe is the most likely place the new Doctor fits in. The scene below takes place at the end of War Games, and marks the end of Patrick Troughton's tenure as the Doctor. As you'll see, his is not a normal regeneration.
The next we see of the Doctor, he staggers out of the TARDIS and collapses, already in the form of Jon Pertwee. I believe that it's here, between Troughton begging not to have this happen to him and Pertwee's arrival on Earth, that Martin's Doctor most easily fits in. One possible piece of evidence was front and centre as soon as she got her memory back, with the Doctor changing into an outfit that was seemingly a precursor to the Third Doctor's 60s-inspired garb.
Earlier than that, during the scene in the cathedral when Ruth took down the Judoon platoon, the moves would have seemed familiar to fans of the classic series. That's because she's actually using Venusian Aikido, which was prominently used by the Third Doctor but hadn't been seen before his run. We're told that it was developed by nuns on Venus and that it's most effective when used by beings with five arms and legs, but we never learn how the Doctor seemingly learned it between regenerations and while a prisoner of the Time Lords.
As mentioned in the clip above, the Time Lords are able to remove information from the Doctor's brain before his exile on Earth. While in the original airing they removed his knowledge of how to work the TARDIS, if Martin did come between these two incarnations then it seems they removed an entire regeneration from his memory too.
Threatened with exile and enforced regeneration, it's entirely feasible that a begging Doctor would agree to almost anything to not face that fate. The Time Lords do call on him in later stories to do things they cannot as he is not bound by their rules of temporal non-interference (notably the Fourth Doctor is enlisted to stop the creation of the Daleks), so they definitely recognize his value in that.
While these details are themselves suppositions, the fact remains that this is the most logical place for a new Doctor to fit in.