As I'm lead to understand, spiritually; the house of Israel is the dominion and the house of Judah is the sanctuary: Psalms 114:2. Allegory of the carnal and the spiritual; that is to say, the flesh and the spirit. The spirit is willing, though the flesh is weak. The joining of the two sticks as one means the spirit and flesh come into unity and an perfect expression of God. Much as the prayer "thy will be done in earth as it is in heaven" means "be done in the outward man, as it is in the inner man": but now we do not see this but: Romans 7:22-24, Romans 7:25, 2 Corinthians 4:16, Ephesians 3:16-17
But about any literal understanding, I'm not sure. It could be America and Britain are the literal offspring of Joseph and the house of Israel in the nations of Europe. There are some intriguing evidences; but in the end, I find it entirely impossible to determine, not only who are the literal bloodline of house Israel, but, who are the literal bloodline of house Judah. I believe God has made this impossible to determine to show the irrelevance of the fleshly seed of Abraham, but instead to focus on the spiritual seed who are the heirs: Romans 9:7-8, Galatians 3:6-7, Galatians 3:27-29