From time to time, we all encounter people who bang on about the Bank of England being independent, or somehow a private sector entity, or not part of government.
Allow me to direct those people to the Bank of England’s website, which very explicitly states that the BoE is a public body answerable to parliament. In terms of government vs non-government, it's definitely part of the government, rather than the private sector. To quote the BoE:
“We are a public body that must answer to the people of the UK through Parliament.” 
Then we come to the “ah, but the government grants the BoE independence to set monetary policy/interest rates)” talking point.
Again, quoting from the BoE’s website:
“The Monetary Policy Committee (MPC) is made up of nine members... appointed directly by the Chancellor.”
"MPC members serve fixed terms, after which they may be either replaced or reappointed.” 
"We set monetary policy to achieve the Government’s target of keeping inflation at 2%.” 
So, the Bank of England’s Monetary Policy Committee, which is hired and fired by the government, has “independence” to set interest rates to achieve the government’s target.
There are many parts of government like this, that have independence within government to deliver the government’s agenda.
The BoE is “independent” in the sense that an individual police officer or a police force is independent, and free to take whatever action they feel necessary, within rules determined by government, to achieve outcomes determined by government (hopefully democratically-decided outcomes).
If it turns out that an individual police officer or a police force can be proven to be, say, (institutionally) racist, most of us generally don’t take the position that the police have been granted a degree of independence to act as they see fit, free from day-to-day political interference, and so the public seeking remedy through their elected representatives would destroy “police independence”.
We rightly see the police as a branch of government and hold it to account as a public body.
Ultimately, both of these public institutions work for, and serve, us, the public.
Hope that helps!