Banking started when cheques were invented in the 1600s when settled payments became a thing. In around 1770 the bankers started meeting in a central place. Then by the 1800s a dedicated space was finally established, this was known as a bankers clearing house.
The Bank of England
The Bank of England was founded in 1694 and was originally a private bank that would act as a banker to the government. It opened for business on the 1st of August 1694. In 1725 the first bank note was produced (as they were hand written before).
Many years later, in 1826, the Bank of England opens its first branch as a response to the financial crisis from 1825 to 1826 and this led to many banks fail. Then in 1894 the first women was employed her name was Janet Hogarth. Six years earlier she had been given a First from Oxford University.
Later in 1994 the Bank of England turned 300 years old! And in 1997 the Bank of England became independent. But the bank is still owned by the government and still operates in the hands of the government.
In 2016 the first polymer bank note was made - the £5 note had a featured Sir Winston Churchill on it.
I found these sources on the Bank of England website.