As a Brit, if we're ranking those things in order of practical importance, I'd put the Royal Family dead last.
People will whinge about the NHS being slow or badly managed, but generally defend its existence to the hilt. Barring some free market extremists, there's no political will in this country to explicitly privatise and dissolve the NHS as a free at point-of-care nationalised healthcare service.
The BBC gets teased for being old-fashioned, quite a bit of stick for the mandatory licence fee, and a lot of well-deserved criticism for the Jimmy Saville mess. Excluding some absolute nutters with dubious motivation and funding, no one's trying to get rid of the BBC.
By contrast, there's a significant political bloc who are at least apathetic towards the monarchy, especially now the late Queen, the last symbol of continuity, has died.