I have the exact same issue. I upgraded my 4S from iOS 5.1.1 yesterday.
I've never had any WiFi problems what so ever - neither with my 3GS, my 4S, my girlfriends 4S or my iPad2 - but as soon as I upgraded to iOS 6.1 all **** began.
I can connect to the WiFi network without any trouble. My 4S with iOS 6.1 works on WiFi for about 2-4 minutes, then it just drops the connection and the wheel at the top left corner just keeps spinning and eventually the website or whatever times out. My iPad2 and my girlfriends 4S with 5.1.1 works fine at the same time this happens to my phone.
So, this is what I've tried without any difference at all;
Phone settings;
- Resetting my network settings, three our four times.
- Rebooting my device
- Resetting my device to factory defaults, totally wiped and clean of anything
- Setting my DHCP setting on phone to static (worked for 2-4 minutes)
- Setting my HTTP proxy to "Auto", instead of "OFF"
Router settings, which of course has the latest (09-2009) firmare;
- Checking UPNP off and on (I had that off before, tried to see if it made any difference)
- Checking that WMM is enabled
- Changing WiFi channel from 6 to 8 (I don't have an automatic option there)
- Changing Channel Width from "Automatic" to both 20M and 40M
- Rebooting the router
None of the above steps helped - I still have 2-4 minutes of WiFi til it drops by itself, and I either have to turn WiFi on/off, or reset the network settings.
This is a big, fat joke.
If anyone has a new thing to tryout, please let me know. Either way, I'll try it a friends network to see if it works there.
If not, and I can't get any good advise from the web, I'll take it back to the shop force them to change it for me, as not everyone seem to have this issue.