What if Tony Blair followed his father's wishes and joined the Conservatives in 1982 and sought to reform the party to become a "socially tolerant, progressive and responsible" party with similar pro-capitalist, pro-competition policies and a low tax system?
If I had the ingenuity and dedication to do a TL, I'd have Neil Kinnock beating the Tories in 1992 after John Major's affair with Edwina Currie being leaked to the press just before the election. However, the party will be hit by economic crisis with the ERM and infighting over financial policy and Europe.
By 1992, Blair becomes Conservative Party in a closely-fought battle and five years later, in 1997, he leads the 'New Tories' to a majority government, which is all but renewed in 2001 with effectively the same number of seats for each party. However, the 2003 Iraq War rejuvenates Labour as the 'true progressives', owing to their anti-war position. However, the Tory majority government proved too strong to overturn in 2005. Nonetheless, it set the story for 2010. Labour replace their leader since '97, Gordon Brown, with the young but aspirational David Miliband, and Blair set off for the States in 2007, himself being replaced by David Cameron.
In 2010, Miliband's Labour Party manage to return to office for the first time since Blair's success of 1997, albeit with the support of the Liberal Democrats in Britain's first peacetime coalition since the 1930s.