I had been having the same problems, a connection test from the internet menu on wii settings were showing fine, the homebrew channel was showing as connection ok but none of apps would initialise network, could not even find game covers for usb loader.
I had assumed it was an issue with my router or connection so I rang my ISP and they advised me to change my wireless channel in my routers settings and check the MTU value in the wiis internet settings and delete the existing connection profile and reconnect from scratch by configuring it manually and it worked now everything works once again, so incase this could help anyone I thought id post step by step instructions
1.from the main menu go to wii settings and the internet tab and delete your existing connection
2.go to manual configuration and make a note of your routers SSID and WAP/WEP security pass(these will normally be on the router itself)
a) manually enter your routers SSID
b)manually enter your WAP/WEP pass
c)check the MTU value mine was 0 for some reason, change it to 1500(nintendo recommended)
the wii will then do a connection test, if successful it will ask you to do a system update DONT DONT DONT !!!!!!
3 go to the homebrew channel and the little globe in the bottom right hand corner should blink and stay lit, if so your connection has initialised
4. go to a homebrew app that uses an internet connection like usb loader and see if it will download a cover or any app that uses a connection but nothing that downloads and installs anything that may brick your wii!!
5. If it works great, if not then using a computer open up your internet browser and in the address bar type in your routers access address to access your routers settings, in there there should be a wireless settings page where you can alter the wireless channel, it may be trial and error but you should find a channel that works.

This is how I have solved my connection problems, im only a noob but it worked so it may be worth a try.