The RBoS Digital Banking site will let you in using O2 if you change its ID string to IE5.5. It's okay on the log-in pages and you can see a mini-statement and download a full statement in CSV format. But trying to enter an amount in its money transfer boxes gives an 'invalid character entered' error. The same is true for RO Firefox.
Egg banking works perfectly with O2; I don't think I've tried it with FF.