I have a bank account with TSB and got compensation as a result of this mix-up. Some rather personal experiences of the fiasco: – Rather pointlessly, the website changed from being mostly static to entirely written in a very JS-heavy, "dynamic" way. I still can't use it in my normal browser (FF) with its extensions because it relies heavily upon CORS requests and referrer information that my somewhat privacy-paranoid…
Paper did not have those incompatibility problems...
However, from the BBC article I conclude that even customers with a default browser could not necessarily use their account
Edit: Forgotten not added.