As you point out it took near enough a solid YEAR for them to produce series 12. They started shooting in January, and were finishing shooting somewhere around October or November. The material would have still being going through the post-production process to add effects etc, and there was some questioning as to why it was it was taking so long to shoot a season that had two less episodes to produce than in the Steven Moffatt/Russell Davies era's...
There were no obvious answers, though one theory I had was that because the budgets had been pruned and pruned perhaps there simply weren't enough hands to actually make it to schedule anymore. Less people means a slower rate of progress. So with that painfully slow Production rate in mind it makes perfect sense that with the Pandemic and its after-effects, shooting series 13 would take... considerably(?) longer to shoot yes. It's just logical.
The potentially interesting thing to consider when you put all of these things together though is that by having such delays now, and the lower budgeting likely to be affecting the speed of production, mounted on the back of continually sliding ratings, the BBC's enthusiasm for the series might have reached a crossroads. Has it become a Problem Child rather than a Timeless Child?
Even though there are contracts with overseas broadcasters to honour I can see the BBC perhaps opting for shorter seasons, or breaking the season into two blocks. It depends on what they can get away with contractually. Because when you assess the ratings trend for series 12 it takes no great intellect to see that series 13 is going to be potentially disasterous. If you are a BBC scheduler, in charge of putting this programme somewhere, where could you put it if it isn't performing on a Sundays anymore, and would have no chance on a Saturday's therefore...?
Someone at the BBC is surely asking these questions.
Could the BBC just decide to pull the plug on a tv show of their DR and move towards a Movie franchise, like when Star trek went from TV to movie screen?