Individual banks existed long before there was a central bank, and the central bank, the Bank of England, was set up in 1694 as a private company with the express purpose of lending money to the state, which had lost all creditworthiness following vast expenditure first by Charles II when he was forced to default on his debts, and subsequently by William of Orange in wars against the French. Many schemes were put forward to solve the state monetary crisis, mostly based on raising funds to be repaid by future taxation, but none could escape the fact that money could only be borrowed at punishingly high interest rates because of widespread and justified distrust. In fact merchants could borrow at significantly better interest than could the state because they were trusted and the state was not. This led to a successful scheme put forward by William Paterson, to found a new bank, to be governed and managed by merchants and thereby reflect their financial credibility, but with the ability to issue bank notes that carried the authority of the Sovereign. This was what Felix Martin has called 'The Great Monetary Settlement' (Martin 2014 Chapter 7). In effect the King enjoyed the credit that only a bank governed by creditworthy merchants could provide, and the bank enjoyed the authority that only the King could provide. The new bank was the Bank of England.
Because of the sovereign authority it suited the separate merchant banks to become customers, because they then had an independent and trustworthy source of money with which to settle accounts between themselves. As a result the BoE's money expanded greatly, lending in turn its authority to an ever widening group of merchant banks, as their money became more widely transferable by their own customers from one merchant bank to another. In a crisis, when people were finding it increasingly difficult to repay their debts, confidence in the merchant banking system was lost because it was the banks that held those debts. At such times the BoE often acted as lender to the troubled banks because it was able to provide sound money by printing it in return for the loan agreements that other banks had accepted and which now no-one else would buy for fear of default. It did this not because it was obliged to do so, but because it could turn a good profit by buying banks' debts at considerably less than their face value, the discount representing interest to the BoE, which was often very high in times of crisis.
To this day banks normally settle debts between themselves using BoE money, although cash is now seldom if ever used for this purpose. Instead they use BoE reserves, which, being electronic, are much more convenient than cash. All banks have accounts at the BoE, as does the government, and these accounts use reserves. The BoE is the banks' and government's banker, and:
Just as bank money never leaves the banks, reserves never leave the BoE.
Debts are not all settled between banks immediately. Most are accumulated during the day and the net difference settled at the end of each settlement period, which varies depending on the type of settlement system used - BACS, CHAPS, cheque clearance etc. Because only net differences are settled the total quantity of reserves at the BoE needed for settlement is only a fraction of the total quantity of bank money in customers' accounts - see chapter 39.
The BoE was nationalised in 1946 by the then Labour Government headed by Clement Attlee.