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.
Gluten free bread recipes are hard to get right, and maybe won't function with kneading per hand (there was a question about non-rising gluten free bread yesterday). You are much better off with an existing gluten-free bread recipe. Pick one which you know to be good, and prepare the dough as usual. Follow the recipe which is described here from the "Take the dough and flatten it out a little" sentence on, using your known good gluten-free dough instead of the one described here.
I don't know what would be the best substitute for the soaked wheat, experiment with soaked grains or myabe something else which gives a slight crunch (sunflower seeds, almond pieces - don't soak these).
Best Answer
Both of those ingredients are not bitter unless they are rancid. You may be using too much baking soda?
A basic gluten free flour is usually a combination of white rice flour, brown rice flour, sweet rice flour, tapioca starch and xanthan or guar gum. From there it can go into buckwheat, bean flours, quinoa, etc. The gums give resilience and elasticity to the dough (along with the tapioca and sweet rice flour) and replace the action of the gluten which creates a structure for baked goods to rise.
There's lots of gluten free flour recipes that come up with a Google search. It's sometimes easier to start with those than trying to create one from scratch.
Oats BTW, are not strictly gluten free because of cross contamination when growing or processing. The package should state that they are gluten free.