The only way to go is to port Mozilla Firebird or a simular popular open software project to Risc-OS.
That way you always will have updates.
Get Opera to launch a Risc-OS version would be great but it's very unlikely they would do that even though they have ARM versions. Another low chance posibility is to get an embedded version of Risc-OS so that the scale of operation (mobiles) gets large enough for Risc-OS browsers to get more development.