Okay, so some news on the possible Season 13 may have leaked from the BBC. Please note that at the moment these are JUST RUMOURS, and should be treated as such unless and until something more concrete surfaces.
However, these new 'rumours' apparently come from the same source who broke the news that Season 12 would retcon the origins of both the Doctor and the Time Lords, and that the Doctor would be revealed to have started life as a female. So they can't be entirely dismissed out of hand.
1) While the BBC are entirely happy with the narrative developments over the last two years, they're extremely concerned with the rapid and severe ratings decline. Discussions have been underway for some time regarding how to reverse the decline, and as such, pretty much every aspect of the show is up in the air right now.
2) Recording of the new series was scheduled to begin in September 2020, but now looks likely to be pushed back to early 2021. (This isn't much of a shocker given the head of BBC Wales' recent statement).
3) This will definitely be the last series for Chris Chibnall, Jodie Whittaker and probably all three companions. Now, we already knew that Bradley Walsh and Tosin Cole were leaving, and it's been strongly rumoured for some time that Chibnall and Whittaker wouldn't be renewing their contracts, but it now seems that the BBC are considering dropping Mandip Gill as well and starting fresh if another series is commissioned.
4) The next round of recordings are likely to contain the same number of episodes (10 regular plus 1 x Special). However, they're considering splitting the season into two parts.
The first part will air either spring 2021 or autumn 2021, with the second part airing either autumn 2021 or early 2022. Possibly all episodes will be held back until 2022. It all depends on when shooting actually starts.
This leaves the placement of the special up in the air. If they decide to air all ten episodes in 2021, then it'll go out on New Year's Day 2022 and be the last story of the Whittaker Doctor.
If they split the series over 2021/22 it'll probably go out January 1st 2022 (Halfway through the series), but it may be held back for transmission on January 1st 2023.
5) They're considering halving the number of episodes per season.
This is potentially a biggie. Basically, what it means is that while all 11 episodes will be shot in one continuous recording block, if they decide to split it into two, the first batch of 5 or 6 will go out as Season 13, and the second batch of 5 or 6 (Depending on where they place the special) will go out as Season 14.
Why is this big news? Because doing it would help the BBC fulfil their contract to HBO Max quicker. After 'Season 14' airs, the BBC would only have to deliver a further 5 new episodes and their obligation regarding Doctor Who is done.
So what happens after 'Season 14'?
It's anybody's guess at the moment. If they want Doctor Who to continue, 5 episodes would be a good testing ground to see if a new Doctor and a new approach is viable.
If they're considering ending the show, then they just have to figure out how to fill 5 episodes without Jodie Whittaker. They MAY go with a brand new Doctor. On the other hand, they may opt to bring back an earlier Doctor. If that's the case, then my money would be on Jo Martin. Just give her the ball and let her run out the time on the clock.
Of course, all of the above is dependent on Doctor Who coming back at all in 2021/22. I'm still half convinced it won't, but these new rumours certainly seem like the BBC are trying to figure out a way of resting or ending the show quietly and inexpensively as possible while fulfilling their contractual obligations.