If A is lying, he is Tweedledum, and it is Monday, Tuesday, or Wednesday.

On those days Tweedledee tells the truth, but B says he lies on Monday, a contradiction.

So A is telling the truth, and he is Tweedledee.

Therefore B is Tweedledum, and he is telling the truth when he says he lies on Mondays.

Since both of them are telling the truth, it is Sunday.