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.
Best Answer
I am not really giving you a recipe, I am give you a technique.
There are two ways to make noodles, you either boil them and then use boiled noodles as an ingredient, or you put in just the right proportion of moisture to dry pasta so it finishes just right in a very similar manner to cooking rice.
If you want noodles that are full of the dish's flavor, then you want to do the second choice.
The noodles are most likely cooked in a chicken broth base or water, and spiced with some blend of powdered or fresh ginger, soy sauce (more when using water less when using broth), teriyaki or hoisin.
When cooking with angel hair (not the same as lo mein thickness and will behave differently), just about 6 cups of chicken broth will be absorbed fully by 1 lb of dry boxed angel hair pasta.
Just to make sure this is clear, the pasta would be the last ingredient added for this pasta base. Of course you can add other things to your broth first like your mushrooms, meat, shoots, onions/leeks, water chestnuts, whatever you like in your lo mien. Add the pasta, stir it and after it has absorbed most of the water but before it starts burning on the bottom start tossing it around the wok to give it that smoky stir fried taste and texture
Well if you like fresh spinach in your lo mien, you would actually add that last and stir it in until it's wilted just before serving.
Fresh pasta will NOT absorb anywhere near as much water so keep that in mind if you plan to make fresh pasta.
Have fun experimenting!