I use my wiiu on a daily basis to watch videos, also on youtube and I dont have to switch user agents... is it because I have a pal wiiu?
but a little trick, that might be usefull for you is to try exchanging the "watch?v=" of a youtube link with “embed/” I use it, so I can watch whole playlists without having to start every video 1 by 1, but maybe it can also help you with your other problem


edit: also try "_popup" instead of "embed/", if it still didnt help you


so for example. if you have a playlist and it looks like that
youtube.com /watch?v=XXXXXXXX&list=YYYYYYY#t=2
then you change it to either
https://www.youtube.com/embed/XXXXXXXX#t=2
or
https://www.youtube.com/watch_popup?v=XXXXXXXX#t=2

and yes, you take out the "&list=YYYYYYY" part of the link or else the wiiu cant load the vid

 

well this "fix" is mostly about having playlists autoplay
dunno, if there is a useragent that enables autoplay, but for me it doesnt automatically activate autoplay on playlists... it plays a video, jumps to the next one and then I have to press "play" again for it to start the next video
this "fix" though lets it play a playlist completely on autoplay *yay* ^^

 

well I dont have to change useragent for youtube either, works just fine, just the autoplay in playlists doesn't work unless I change the beforementioned part in the URL, no idea, why WiiCube_2013 has to change the useragent....
whats your wiiu's system software and region? mine is 5.0 pal


edit: maybe if you try accessing youtube over this link https://www.youtube.com/html5 and change the video player manually, then you dont have to switchthe user agent....

 

My Wii U is PAL (UK) and running on the latest software update and it's been like this for ages, maybe cleaning the cache or something would help.