The Youtube app is now available as a free download on the Nintendo eShop for all 3DS systems around the world. It's actually a fairly well done app, and it uses the 3DS's two screen structure really well to perform multiple functions at once. It works fast with no lag, as well.
Oh, and to make this more System Wars worthy: lol, Google will put its apps on 3DS (and Vita even), but it won't put them on Windows Phone. Lol Microsoft. Just, lol.