Trust me, New Labour (that be who you must mean with the Fabians right, they are a Labour Party think tank) were not socialist. And we have a Coatlition government now which is Tory in all but name that is anti-Fabian. In the last few decades we've had privatisation of almost all nationalised institutions (railways, water apart from in Scotland, energy, steel, motoring, telecommunications) and the huge growth of a financial sector. and the following banking criis, and growth of the super rich. Healthcare is the last big nationalised institution and hell Cameron is trying to dismantle that too now, and certain aspects of the bloated welfare state, which I agree with him.
A socialist British government wouldn't have abolished the 10p income tax rate for the poorest in society, nor would it be cutting funding to unviersites by 80%. We had under the labour government some social policies that could be interpreted as socialist but the current government is trying to rebalance the structural deficit by pruning the very large public sector, creating all the protest marches you'd expect