It is a total mess but the US President being assassinated by the Master was ret-conned when the time paradox ended. It never happened.
No, actually, that still happened. Right after the reset, we hear someone over a radio demanding an explanation saying "we just saw the President assassinated."
It's the problem with trying to keep it real in an ongoing story. Fine to start with but you play out one story it's hard to pretend they are still in your universe.
007 would be even harder to explain (so they don't bother!) Each Bond is essentially a reboot, yet with actors carrying over from the previous incarnation in the same role.
If you look at Winters in The Sound Of Drums, even as a straight Bush Jr substitute, that changes the course of US politics for the Whoniverse. Yet it then seems back on course with Obama, by The End Of Time.
But potentially a future Doctor Who story could visit the White House in 2007, with George Bush Jr. as a historical figure.
UK Policitics is even more muddled!
Doctor Who's depiction of Earth politics is confused to say the least. Its pretty clear that the writers know nothing about how politics, event in the UK (but especially internationally) work.
The Winters thing was messed up on a number of levels and apparently RTD admitted that he did not fully understand the American Presidency (to which I say to him...watch a few eps of the West Wing). The Sound of Drums takes place in the spring of 2008. The US would not have an actual "President-elect" until the late fall. We can only conclude that Obama was Winters VP.
Of course this is the same TV series that stated conclusively that the UK needed permission and codes from the United Nations in order to fire its OWN nuclear weapons.

More astoundinly...Winters states that the UN could have had Prime Minister Saxon REMOVED FROM OFFICE

Its probably stuff like this that made the UN leery of being associated with Doctor Who,