I seem to remember something about her being the most senior MP by the end of World War 3, but I may be wrong.Apart from the fact she apparently took control of the party and government.
The Doctor asks her if she won the election in "The Christmas Invasion," and she replies that she had a landslide victory.
So the fact that she later becomes Prime Minister is evidence of nothing other than that an election was called.
I don't remember any such thing. She was the only MP present in Downing Street during the Slitheen infiltration, and the Slitheen impersonating MP Joseph Green, the Acting PM, had ordered the planes that were carrying the Cabinet members back to London grounded in order to keep the Cabinet out of things.
After Number Ten was destroyed, Harriet Jones took charge of informing the public that the crisis was over, but that was as far as her authority was established to go -- completely unofficial until the election that made her PM.