Both BBC1 and ITV1 did some rolling Royal Birth coverage in the 2010's and certainly one did fall on a Saturday early evening - Princess Charlotte. Prince George coverage fell in weekday primetime, both the birth announcement and leaving hospital IIRC. Can't recall if anything of significance got replaced, or whether it was a case of only displacing repeats. Prince George as well may have been treated differently due to being directly in the line of succession
Yes, Prince George did get rolling coverage - it broke at 8.30 on a Monday night so BBC1 dropped Panorama and the 9pm show and rolled until ten o'clock, though as you say that was a bit different because he is directly in line to the throne so it was a bit more important constitutionally. When Charlotte was born, it was on a Saturday morning - I remember it being announced by Darren Fletcher on Fletch and Sav on BT Sport - but there wasn't rolling coverage for that, they did a little newsflash during Saturday Kitchen (though presumably that was just because there was a live show on at the time, if there was something pre-recorded on they probably wouldn't have) and then I think later in the afternoon the snooker underran so they did an additional news bulletin.
I remember one time there was potential for major news (can't remember if it was a royal birth or one of the many occasions Nelson Mandela was rumoured to be gravely ill) fell on a very busy Saturday night so various contingencies were drawn up for what might happen if the news was announced at different times. But of course with most major news stories there's no possibility of forward planning so these decisions need to be made on the fly - and like with a lot of things just because one decision is made on one particular occasion, it doesn't mean the same would be done on another.
Yes, and Mandela's death was of course a famous one because it broke very soon before the Ten O'Clock News and there were lots of complaints that the Ten was devoted to nothing but, on a night of exceptionally bad weather across the UK. But I think that was a good example of the best-laid plans going awry, because I seem to remember that it had been said before that the plan was that they wouldn't announce his death after dark and if he died during the night it would be embargoed until the next morning. But that seemingly didn't happen and they announced it at about 9.50pm, which meant all the tributes were coming in when the Ten was on air, so whatever balanced programme they'd planned had to go out of the window.