I've used 'Bon Echo' builds to access NatWest on other platforms before. I suspect Mozilla might allow the UserAgent to contain 'Firefox' for this reason. Perhaps this could be fixed for future RISC OS Firefox builds?
On another note: secure sites (banks in particular) use this method as they see it as a least-worst approach to protecting their customers from fraud. It shouldn't really be used at all though.
Banks claim to test each major browser version for immediate security holes (mainly phishing/malware I suspect) before allowing access to their site. I wouldn't be surprised if some banks exclude IE6. Certainly IE5 and NS4 wouldn't be permitted now, as they no longer receive security updates.
Provided the bank can test a given browser, they have no excuse not to add it to their white list.