If it is Tuesday, then B is lying, so B is Tweedledum.

But that means A is Tweedledee, and is telling the truth when he says Tweedledum will tell the truth tomorrow, a contradiction.

Thus it is not Tuesday, and B is telling the truth.

It can not be Sunday, because then A could not be telling the truth.

Thus A is lying, which means Tweedledum will lie tomorrow, which means it must be Sunday, Monday, or Tuesday.

Since it is not Sunday or Tuesday, it is Monday, which means A is Tweedledum, and B is Tweedledee.