The order, of course doesn't matter, either in Korean or English. Today, nobody has played with the dog. or Nobody has played with the dog mean the same thing. 아무도 오늘은 or 오늘은 아무도 also mean the same thing. I am not a native Korean speaker, but I think 누구도 is also ok.
The main point of the sentence - present perfect tense-- is being missed as far as the study of the English language. The Korean sentence could also simply be "Nobody played with the dog today." By inserting "has" we know the day is not over, so there is still time or opportunity to play with the dog.
In Korean, we might add something at the beginning such as "아직" or "이때껏"