It doesn't quite work like that.
The staff in the branch can report someone to their head office if there is suspicious activity on an account, they then decide if it warrants police involvement, it comes under money laundering regulations.
Also all banks now have Anti Money Laundering (AML) software that scans accounts and looks for "possible suspicious" behaviour which includes paying in more than a certain amount in cash or multiple smaller amounts of cash. For example, where the account holder is flagged as a student, paying in cash multiple times could automatically raise an alert as would say paying in more than £10k. If the account holder is flagged as a "taxi driver" then it would be expected that he could often pay in cash.