It's been a while since I've seen that episode, but how was overthrowing Harriet Jones changing history? Was there some original timeline where she remained Prime Minister for a longer time? I just assumed that the Doctor had always been the one to remove her from office.
Nope. As others have said, Harriet was originally supposed to serve three terms as Prime Minister and usher in Britain's "new golden age." Not only did the Doctor change history by getting her tossed out on a vote of no confidence, he opened up the door for the Master to take office as Harold Saxon and for the corrupt Brian Green (seen in
Torchwood: Children of Earth) to take office.
I don't buy into his "Time Lordy senses." Even the Tenth Doctor admitted that his "fixed moment in time" theory was in fact just a theory. He had no proof.
He only called that "just a theory" when he was trying to pass himself off as a human from the "present" to Captain Adelaide Brook in "The Waters of Mars" -- he was trying to present what he knew to be true without tipping her off yet that he was more than just another Human. It's not a theory, it's a fact; it's how he perceives time. He is able to recognize which moments in time must stand and which are alterable. He repeats that in "Cold Blood," in point of fact.
Plus, I feel like it completely negates everything he learned in "The Waters of Mars." He can't just change history because he feels like it. "The Time Lord Victorious is wrong."
The Time Lord Victorious is wrong because he was trying to alter a fixed point -- he was trying to change the death of Adelaide, which
had to happen. That was what was wrong -- he was trying to change the fundamental structure of time, an essential event that cannot change without all of history being screwed up.
To put it another way:
The Doctor changes history all the time. Every time he gets involved in historical events, he's altering history. When River was about to die at the end of "Forest of the Dead," he yelled out, "Time can be re-written!" only to have her refuse to let him re-write her life with him. The Doctor can change your life history without fucking up history most of the time.
What causes giant bats to appear and eat people, what causes time itself to fight him, is when he tries to fuck with the Fixed Points. It was his hubris in thinking he could control time to the point of changing the Fixed Points that was the issue, not changing history in general.