The NatWest system (nwolb.com) has now been updated and claims to support Apple Safari 1.3 and 2.0 for the first time. I checked RBoS as well and their system hadn't yet changed to reflect this, though a news update saying "We are making the service easier to use and more accessible for customers with disabilities. We will be launching this new service..." suggests it will happen soon.
I've been using the NatWest site with Firefox on the PC with no problems, so today I decided to give it a try with FireFox beta 3 on RISC OS. However, a certificate warning came up, so I chose to accept the certificate temporarily, and on making it through, the NatWest page stated "The Internet Browser you are using is not supported by OnLine Banking. Please use a recent version of IE, Netscape, Mozilla, Firefox, Safari or AOL"
Maybe the Firefox beta is identifying itself as something else, possibly "Deer Park"?