I don’t think to leave EU would change anything if someone is already here legally and still has permission to stay.
There are already rules & laws to prevent people getting bank accounts which are not here legally and Home Office can also close any account if they ‘suspect’ someone is in breach of immigration rules.
Just to clarify, the immigration laws that require you to check the immigration database only apply to those that are known to the Home Office to be illegals. Those who have left the UK, without having breached the terms of their leaves will not be on that databases and banks are not obligated to close their accounts.
(1)A bank or building society (B) must not open a current account for a person § who is within subsection (2) unless—
(a)B has carried out a status check which indicates that P is not a disqualified person, or
(b)at the time when the account is opened B is unable, because of circumstances that cannot reasonably be regarded as within its control, to carry out a status check in relation to P.
(2)A person is within this subsection if he or she—
(a)is in the United Kingdom, and
(b)requires leave to enter or remain in the United Kingdom but does not have it.
Just a quick note on this and I am happy to be corrected. As far as I am aware, the European directive on basic accounts (which doesn’t restrict which EU country you can open the UK account in) only legally applies to nine UK banks.
0.6 Designation of credit institutions to offer basic bank accounts
At Budget 2016, the government designated the nine largest providers of current accounts in the UK under the Payment Accounts Regulations. This means that the nine banks are legally required to offer basic bank accounts to eligible customers from 18 September 2016.
The nine banks are:
Barclays
Clydesdale and Yorkshire Bank
Co-operative Bank
HSBC
Lloyds Banking Group (including Halifax and Bank of Scotland brands)
Nationwide
Royal Bank of Scotland (including NatWest and Ulster Bank brands)