Looks like it's just the Post Office for now. I found this on the FCA's website that alleges money laundering is more common via the PO than directly through bank branches:

https://www.fca.org.uk/firms/financial-crime/cash-based-money-laundering

No doubt this is more likely because bank branches are rapidly vanishing all over the country than some fault with the PO itself...

As seems to be the British MO these days, it's easier to inconvenience everyone than investigating, preventing, and punishing criminal behaviour. Case-in-point, Right to Work, Right to Abode, Voter ID, etc., etc.