I don't think they have the regional capabilities on BBC2, just the nations. I could be wrong though.
Digital BBC Two indeed was never designed to be able to split into the umpteen English Regions AFAIK, like its analogue predecessor could.
That's why the regional political and documentary shows (i.e. what is currently Sunday Politics and Inside Out) moved to BBC One, ahead of Analogue Switch-Off.
Before those programmes moved channels, viewers in all regions watching digital BBC Two would see e.g. LDN mid-afternoon bulletin on weekdays, and Metropol on Sunday lunchtimes. I think that digital BBC Two might've replaced the regional doumentary strands (which each had unique branding) with something that featured a selection of items from about 3 regions per episode (possibly branded as "Homeground"?).
There was definitely one occasion when a weekday edition of Breakfast was shunted to BBC Two (by sport, presumably?), so digital viewers saw the main Breakfast presenters do extra mini "newspaper front pages" items in place of each regional bulletin.
Nowadays the Wimbledon effect (in years when there isn't a simultaneous World Cup, at least) knocks regional news to e.g. 7pm or later on BBC One, rather than moving channels.