The Bank of England is wholly owned by the UK government, after being nationalised shortly after the Second World War. It isn’t a private company.
There are often questions about a subsidiary of the Bank of England, known as the “The Bank of England Nominees”. This subsidiary is owned in whole by the Bank of England and one nominee shareholder, who is also an employee of the Bank of England. As this Hansard record states, it holds shares on behalf of ‘Heads of State’ and the government. It has less than £1million in assets that it owns itself, as the Bank of England annual report states, which means the company itself doesn’t hold very much at all – it just holds securities in custody for other groups. The Nominees has now been liquidated and no longer functions.