Using without a SIM

This is NOTHING to do with activation or anything to do with the cellular network.

For over a year now I've been using a couple of older iPhones just making use of their WiFi connection to my local network and hence the Internet. The other day I noticed the 3Gs could not make use of the network. It sees the correct WiFi network and apparently connects fine (no complaints), but then nothing else works over the network. At its most basic, Safari cannot even access the local web server, so it's nothing to do with any Internet connection (or lack of).

I've tried with Airplane Mode on and off, with SIM and without, turned WiFi on and off, restarted the entire phone, everything I can think of, but I just cannot get the network to do anything other than connect.

It's not the iPhone that's faulty in some way as I tried the other iPhone (4) in the same situation and lo and behold, that now doesn't work either, although an iPhone 5 with working SIM functions perfectly on the same network. So from this I deduce it's an iOS thing rather than hardware. Did it start with iOS 6.1.3? No idea since that's some time ago and I cannot determine exactly when this problem originally surfaced.

This is a new problem as these iPhones have been doing this perfectly for a long time, now suddenly, both cannot use the network. Has anyone else seen this and/or got any suggestions?