In answer to my own post, I rang the Roberts technical support hotline and spoke to a pleasant but completely useless lady who suggested helpful things like rebooting the router and the radio. Is it me?
On the basis that I have two identical radios, both of which worked perfectly well before the recent software upgrade and neither of which worked properly after it, my suspicions lay in that direction, so I played about a bit. And sorted it!
Neither radio works with the internet if the IP is provided by my router's DHCP server and the fix is to set up a static (fixed) IP address on the radios. This is weird, as the only real difference is the way the radio resolves addresses from DNS servers. Oh well... it's not my problem, so if Roberts are reading this then could they take a good hard look in that direction please? Ta.
If you're going to do this fix then you'll need to know the SSID that your router is doling out, and ideally the pool of addresses that your router's DHCP server doles out. You can usually get these, and your gateway address, from your router's web access page. If you have the same problem that I had, then you can also see this info by connecting your radio to the internet - even though I couldn't access radio stations I could at least see the connection settings on the radio.
Then set things up. My router's addresses start with a 192.168.1 - not all routers are like this, and yours may start with 192.168.0 or something else, so pretty obviously tweak the stuff below to match your router.
Enter the manual network setup area:
Menu => System Settings => Network => Manual Settings => Wireless => DHCP disable
This is the IP address I chose. It's above the highest address that the router's DHCP server can give out (192.168.1.240), so there's no chance of an IP clash with other kit around the house.
Enter IP Address: 192.168.1.241
Standard subnet mask follows:
Enter Subnet Mask: 255.255.255.0
Again, a pretty standard gateway addy:
Enter Gateway Address: 192.168.1.1
Let's use Google's free dns servers:
Enter Primary DNS: 126.96.36.199
Enter Secondary DNS: 188.8.131.52
Fill in your SSID (this is the network name your router pushes out)
Fill in your wifi authentication method and key.
Job done and everything now works again properly. Hope this helps someone (possibly Roberts?)