Knowing that Jodie has already filmed her part of the regeneration scene, and the more recent news that they've disassembled the TARDIS set (sorry, not sorry), suggests to me that she'll:
A) Regenerate a la Troughton and her successor will stumble out of the TARDIS in her garb, or a messed up version of it. This also worked in a 1970 Doctor Who stage play where the actor portraying the Doctor was meant to pick up after Pertwee.
B) She regenerates outside of the TARDIS (a rarity for the current production era!) and the new person stands up and saunters off to their next adventure, finding a new TARDIS set once they find the blue box.
C) There is a continuity break and the new Doctor is simply seen at the beginning of their next adventure, with the backstory filled in later.
IMO, there are some production realities that would have to be accounted for. Not the least of which is that Jodie is not overly tall statured and if her replacement is male (or otherwise taller - Jodie is 5'4" sans footwear and all the other modern Doctors have been in the six foot range), odds are they'd look more than unusually awkward running around in her old clothes before finding a replacement kit. Not impossible by any means, but knowing RTD I'd not be at all surprised if he started his return with the new Doctor in medias res, leaving only visual clues and a verbal reference to being shorter and blonde when they last looked in the mirror.
Mark
I mean Jodie's trousers are already only 3/4 length so they'll probably look like shorts on anyone taller

Maybe her clothes will tear Hulk style, or perhaps she won't regenerate in her costume but something else that more easily facilitates the change. I'm pretty sure Capaldi's outfit seemed to fit her rather than falling off her.
The Tardis set could be easily reassembled, you might not even need all of it, just a portion and green screen the rest.
Actually just rewatched Eccleston to Tennant, it's amazing how little of the Tardis you can see behind Tennant because it's mostly done in closeup. Depending where/how Whittaker regenerates I think it will probably be incredibly easy to insert the new Doctor. As I said upthread it seems unlikely Dan or Yaz, or anyone else, will be with her.
Nah, this isn't a doom scenario. I'll eat my entire library if the announcement isn't directly tied to the BBC Centenary celebrations.
If they're auditioning now I expect them to know who they want within the next month or two at most. Keeping that secret until November might be tricky. And for people who say the audition process has only just started, well Tennant announced he was leaving in October 2008 and they announced Matt in January and Moffat always implied they saw a lot of people.
You library is safe. There will even be an interview where RTD talks about how he loved it being a proper surprise. (Or it will just be a total reboot — more or less like it sort of was in 2005, until the history is dripped in again… it wasn’t until Human Nature we had confirmation that McGann in particular was ‘canon’ and am fuzzy if the ‘classic’ show was ever hard referenced until Sarah Jane showed up.)
Yeah but the situation is different now. In 2005--outside of the TVM-- it'd been 15 years since Who ended, people had forgotten/never watched it, it was a blank slate, we're talking a new series that will not only air a year after the previous series ended, but one that has to celebrate the 60th anniversary of the character. Everyone knows what Doctor Who is these days, even if they don't watch it, it isn't that creaky show mum and dad used to watch. I still fully expect a soft reboot (though hopefully not as soft as series 11) but I can't see it being quite the blank slate it was in 2005.
Sorry, I don’t understand why the next Doctor must be cast.
Why is it not just that unnumbered Doctor from the Timeless Child reveal? I liked her outfit.
I love Jo Martin's portrayal of the Doctor, but bringing her back would involve serious narrative gymnastics.
Whittaker could regenerate into a new version of Martin, the Doctor has used familiar faces before (Colin, Capaldi, the Curator) but is it feasible that Martin Mk 2 would have the exact same outfit/personality?
If you just start the new show with Martin already the Doctor, then either she's working for Division (ugh) or she's on the run from Division and hiding out as a human in 21st Century Gloucestershire. Neither idea appeals.
The only other option is some convoluted idea whereby Martin was 14 all along and much as people would love the Timeless Child untangled, that'd be a nightmare to work out.