The Sun backed Labour both in the '97 and '05 elections, so its wrong to think of it as being unquestionably 'right'.
I think you misunderstand what was going on in British politics at that time. The Tory government, by doggedly following some pretty disastrous policies in the early 90s (Poll Tax anyone?) had lost almost all its credibility with the public. They were clearly going to lose the 1997 election, big time. The ruling elite needed to prevent a genuinely leftist government from coming to power, which meant that the Tory Party had to be thrown out the window and a replacement found to fill the vacuum. The 'New Labour' Party stepped into the breach in the nick of time, and continued the Tory Party's policies under a new trading name and new management. It is very telling that the first public figure Tony Balir invited into 10 Downing Street for a long chat over tea and cakes after his landslide victory in 1997 was Margaret Thatcher. He was basically telling her, "your legacy is safe with us." This is the context for Murdoch's news media switching their allegiance from the Tories to New Labour. It was not a move to the 'Left' at all, but a continuation of its Rightist stance by other means.