Ooooh, Phyl, I can tell you're in for some frustration. Some times (and I mean no disrespect to anyone or any breed of any size) the little guys can be more difficult to house train than the big ones.
Is there any pattern to when he goes? It might help if you made him come and ask for water when you're home with him, then watch him to see how long it takes him to have to go. Corgan's big enough that you can set a water dish for him that will be out of Nathan's reach. Leave Nathan's empty water dish on the floor so that he can go to it and scratch or bark or do whatever does to let you know he wants something. This will make it easier and less traumatic for both of you, and way less confusing for Nathan. Once you start to get an idea of how long it takes his water to run through him, you'll have a head start on catching him and taking him out before he gets a chance to have an accident. Then you can do the standard things, praising him, petting him, etc. It may take awhile, or he may pick up on the idea pretty quickly once you're able to catch him in the act a few times.
One word of caution, though, make sure you give Corgan attention when he goes. It wouldn't be unheard of for him to get jealous and start using the house again in order to get attention!