Central bank independence is a policy goal of neoliberal economics that seeks to undermine democratic control of the economy and the accountability of the government for it, but it is not what happens in practice.
In the evidence given to support this it is said that:
What this makes clear is that the purpose of the move was political: it was to reassure markets that a new Labour government would be seeking to prudent with regard to government finances by passing control of interest rates to the Bank of England.
Quite, the point of central bank independence is to take control of interest rates and the money supply away from whatever nutter manages to get elected. Or at least to make it more difficult for whoever does to Zimbabwefie the economy. Which is why it was brought in by a Labour government. They being at most risk of the belief that they will, they therefore gaining the most from the perceived reduction of that risk.