As to why the final daytime BBC2 bulletin was at the peculiar time of 3:50pm, I wonder if this was a legacy of the previous even-more-oddly timed 3:53 regional bulletin on BBC1 in most of England (except London/SE) in the 70s and early 80s.
I think that was the case, the 3.50 regional bulletin was certainly longer than the other ones, and I know in the North West it was read by one of the main presenters, so presumably it was indeed because that was the established slot for regional news - of course at the same time ITV had regional news at 3.20.
In Paul Armstrong's book about editing Match of the Day, he says his first job at the Beeb was at BBC North East and pretty soon after he started he was directing that 3.50 news bulletin. He says that it was quite easy to direct when Mike Neville was presenting because during the last report he'd always say "How long for the weather?" and depending on the time remaining he'd either wax lyrical about it at some length ("it'll be a nice day for exploring the ramparts at Alnwick Castle, but watch out if you're hoping to enjoy an ice cream on the seafront at Whitby") or just dash it off a few seconds ("Minimum 4, maximum 6. Goodbye.").
I remember, from the school holidays, that the other daytime bulletin that was a bit different from the others was the 11am bulletin, because it was followed by an extended weather forecast for mainland Europe.
12.55pm was always the main lunchtime regional opt out, I can't remember when they shifted it to 1.30pm.
It was in February 1996, when there was quite a radical revamp of the daytime schedules. Breakfast News Extra was introduced, almost certainly because that meant BBC1 could now start shows at 9.20 and get one jump ahead of ITV who began at 9.25, and the 10am bulletin was axed. I remember reading in Broadcast at the time that this was considered absolutely amazing because under John Birt you usually introduced more news bulletins, not axed them.
Also as part of that revamp the lunchtime regional news was moved to 1.30, with Neighbours moving to 1.40 and then Pebble Mill at two for its last few months. Then of course there was an even more radical revamp in September 1996 when Anne and Nick and Pebble Mill were both axed.
Or is that example above with Richard Whitmore the same studio just a different angle, and lighter as its daytime?
You'd think it was the same set-up, but I don't know because in the One Day In The Life Of Television book they talk at great length about the revamped Nine, which had started the previous night, and both Michael Beurk and Martyn Lewis say that they were disappointed with how the newsroom set looks on screen and how the shots are so boring ("What happened to those bold shots we piloted?"), when you'd assume that wouldn't have been an issue had they used the same set for the previous two years. So I dunno.