Man, my Wii has trouble running youtube or even videos from the Nintendo Channel. Maybe it's my internet connection.....but I can watch HD on my 360 almost instantly! It's a cool feature but I want to see how well it works.
If you use wireless, move your modem closer to your Wii unit. It worked for me.
Failing that, you could always get a LAN adapter. I eventually did this anyway because our wireless gets simultaneous use in the house and this made sure that didn't affect me while using the Wii.
Of course if you have a 360 it probably doesn't matter that much to you, but iPlayer/Nintendo Channel/Opera (although they really need a browser update) work perfectly for me now.