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.