Sorry to hear of the issue.
Just a note that we don't give legal advice as we are not a law firm (we are an internet forum).
Only a law firm can assess your case if you feel that your data rights have been breached.
Bank details may be sensitive data and not supposed to be sent to third parties without your authorisation. If you had a loss due to the breach, you could indeed claim damages.
The following link gives a useful guide as to figures : https://www.forbessolicitors.co.uk/personal/data-breach-claims.htm
If it's your own solicitor who did this, you could make a complaint to them and then ask the Legal Ombudsman to step in.
Any breaches need to be reported to the Information Commissioner's Office (ICO) however - by telephone on 0303(###) ###-#### They also have a "live chat" facility on their site - you can visit it here: https://ico.org.uk/make-a-complaint/
The ICO can levy a large fine if a breach has occurred - you should report this to them in any event.
As for damages, it depends on each case but it could be between £3,000 and £7,000 for bank / financial data breaches - the courts have not set exact brackets yet as the GDPR law is still fairly new.
You should consider a no win no fee law firm - you would pay nothing up front and you are also protected from legal costs if you lose.
Here are a few you could try :