The bank is not woke its performative BS. They are investing in oil and wepons and the sex trade and any other money making schemes which can possibly turn a profit. And will allow families to become homeless as they recover bank assets / loan repayment.
They can't lawfully mandate the pronouns, but if senior management are prepared to back the ideology by publically telling customers to fuck off you can take it as a given that it will be seen as mandatory by "teambuilding" career minded managers.
They are a bank in a regulated industry which expects that employees roles will be strictly documented and demarked with clear boundries and specific delegated authority.
Following rules and being seen to do so is very important, if you dont fit in box A you are never going to be promoted.
Uk law recognises the power imbalance between employer and employee. The employee would have to surrender their right to privacy in exchange for pay. If not unlawfully the bank could collect the data for the PC sex and the PC gender reassignment and publish it without permission.
At this stage i am beginning to wonder if someone has made off with a large chunk of dosh and the management are stoking pronoun PR noise in the hope that journalists miss the real story.