The George House tune looks as though it is in C Mixolydian with a last phrase that would be more naturally thought of as being in F major. I can't listen to it here so I guess the question would be whether both of those modes are established firmly enough to create a real sense of a shift, or whether the entire thing is simply ambiguous. Either way the two modes use the same notes.

I can only think of one British folk song that is properly bi-modal, as in using two different scales containing different notes. It is in Motherwell and is called 'William Guiseman'. Bronson reproduces it as a variant of William Glenn / Captain Glen but I don't think it has much in common with that song.