The number of factual inaccuracies in the Sunday Times story (which sparked all this off) indicates that F1 has not been axed by the BBC. It is under consideration, but so is everything else in the "sport and children's" department.
The BBC has a mandate, but it's quite vaguely-worded. Each of the BBC's channels also has an official remit that it must meet in order to be licensed.
In the BBC as a whole, F1 is considered a valid choice of sport because it is a global event that brings the country together - hitting both "global" and "community" purposes. (The other purposes are citizenship, learning, creativity and digital).
BBC One (which is where live F1 is primarily screened) has a remit to be the most popular mixed-genre service with many different good programmes. It's supposed to have as much original and international programming as possible. F1 meets all of the above, so that's the channel BBC needs to use for it.
It doesn't help that the BBC didn't precisely plan to take the F1 rights in the first place; it traded them for some underperforming football rights when ITV decided F1 was too expensive for it. As such, the BBC wasn't expecting it to do a whole lot, but it's exceeded the broadcaster's expectations in each of its measurements and was one of its most successful sporting rights properties according to the last review (done in 2008/2009, so the review would have included such things as the Olympics and Euro football championships). The only other sport to hit all its targets with the BBC was Wimbledon.
F1, the Premier League, the Olympics, Wimbledon and the Six Nations are 70% of the BBC's sporting spend between them. That amount cannot increase because the licence fee remained the same.