I worked for 2 major high street Banks, for 27 years.
He absolutely CAN access all of your data. He will be able to see your balance, all of your debit card transactions, your direct debits, standing orders and your address and salary (if it goes into said account), plus any linked credit cards and mortgages etc. It is very easy to do so.
It is NOT audited whatsoever, so I've no idea why a PP said that it was. And it is not a sackable offence. It's an ordinary run of the mill, daily thing to do.
So yeah, if you don't want him to have access, you will need to change your Bank.
PP who said that Bank staff can access the data of customers at other Banks, is talking absolute bollocks! What you can do, is call another Bank and ask them (about a customer), that if you were to present a cheque today, would there be enough money in a customers account to cover it, and they will tell you Yes or No, but that's it!
And you have to phrase it formally, ie. "If a cheque was in your hands today and in order, for Ihastings account, would this be paid" and they would say "yes" or "no, it would be returned RDPR" (return to drawer please represent) if not enough money in account or RD (if not enough money, and never likely to be , ie. don't represent this cheque, it's unlikely to ever be paid.