They're public bankers. It's an arms-length institution wholly owned by the UK government, which has mandated them to set monetary policy to meet statistical targets as a means of de-politicising monetary policy.
It works quite well, certainly better than when it was directly run and the UK had a lot of inflationary instability.
The BoE provides political air cover for political decisions. In no real sense is it politically independent. It can be relied on to do absolutely nothing to promote economic policies that could benefit the majority of the population at the expense of bankers, land owners, rentiers, and speculators.
The bank itself is owned by the government but has independence from it in certain matters of policy. There are certainly many employees who come from private banking backgrounds but the bank is not a private bank.
It works quite well, certainly better than when it was directly run and the UK had a lot of inflationary instability.