The Bank of England has been wholly owned by the government since 1946. It's very hard for any organisation to be independent of the government that owns it. The Bank of England definitely is not. The government granted the Bank of England some independence in 1997, but kept a right of veto over all it does, which means that the independence is pretty limited in practice. The only real tool available to the Bank of England is
quantitative easing, and the Bank only undertakes this with the specific permission of the Treasury, who underwrites all its risk when engaging in quantitative easing activities. In that case to claim that the Bank of England is independent really does make little sense. The Bank of England is independent of the government so long as it does what the government wants. The Bank of England Act of 1998 provides a Chancellor with the option of overruling the decisions of the Bank of England. That means that the Bank simply has a veneer of independence.
Furthermore, the government has the power to remove that veneer of independence by revoking the BOE Act.