What would be a good substitute for rice wine

Try mixing hoisin or miso into low-sodium soy sauce.

From one of my favorite bloggers, Smitten Kitchen, "I often see low-sodium soy sauce suggested as an alternative but I’m not convinced it’s a fair swap. There’s something more caramelized and fermented in the fish sauce that you’d miss. If you feel like playing around, I might whisk some additional hoisin or even miso into that soy sauce for a more complex flavor."

I hate to say it, but I'd be willing to bet that gluten-free udon is about as practical as gluten-free seitan - the gluten is precisely what gives udon noodles the texture that makes them so special.

Having said that, I've bought frozen udon noodles that had tapioca starch as an ingredient in addition to wheat flour, and those were some of the best udon noodles I've eaten.

The other catch is that making udon noodles at home is hard - traditional recipes call for putting the very tough, springy dough in a plastic bag and stomping on it for a few minutes, as the only practical way of kneading it.

This site describes how to make udon noodles with tapioca - you could try replacing its flour with the gluten-free blend described here. I would also consider adding xanthan gum. If you're feeling adventurous, This e-book (not free, but looks pretty solid) discusses a wide variety of gluten-free flours and their properties, and might help you choose flours to use with the properties you want.

If gluten-free udon turns out to be impractical, I recommend soba noodles as an alternative. Those can much more easily be made gluten-free (although most commercially made ones available in the US do have wheat simply because it's cheap and bland) and can usually be used in the same dishes as udon. They have their own distinct texture, very different from udon, and a deliciously hearty flavor.