Back in 2003, the CBBC, the Britkid's channel, presented a satirical sitcom called, 'My Dad's The Prime Minister'. It started out to be a show for younger viewers, but it caught on with their parents as well and was moved to a family-viewing hour.
Here's how the BBC website describes it:
"Pity poor Dillon Phillips, the prime minister's 12-year-old lad. His status as the son of the country's leader generates major problems: he's a source of scorn and bullying at school, his parents are always busy (his mother is an executive for an international pension fund corporation and remains singularly unimpressed by her husband's position) and life's everyday routines are complicated by elaborate security procedures.
Worst of all is Duncan Packer, his father's smarmy spin doctor, whose only interest is the PM's standing. The omnipresent Packer will stoop to any level to keep the PM in power, regardless of what it may mean to his family, Dillon especially."
It's not hard to imagine that Dillon's father, Prime Minister Michael Phillips, could have been a successor to Jim Hacker of 'Yes, Minister' and 'Yes, Prime Minister'. In fact, there could have been several Prime Ministers between both of their administrations.
And all three shows would exist in the same alternate TV dimension that housed American counterparts 'Hail To The Chief' and 'Mr. President'.
I really should read Ryan's website about British TV shows more often; not that I would have been clued in to see this show in time, considering it's never made it over here to America (so far as I can tell).