Sauron's war on Middle Earth is somewhat more complicated than indicated in the movies. Sauron does not just attack Minis Tirith. His war takes place on three fronts, a familiar concept to a writer who had seen both World War I and II.
Excluding the war on Rohan (which is effectively won by the "Allies" by the end of The Two Towers), Sauron attacks Minis Tirith, Lothlorien & Mirkwood, and the country of Erebor (which includes Dale). The latter "front" is why a prescient Gandalf is so eager to oust Smaug in The Hobbit.
Tolkien discusses the three fronts of the war in his "appendices". And the novels hint at some of the events. However, Tolkien utilizes his third-person narration with great caution, only occasionally breaking the current viewpoint to inform us of events in other locations. Otherwise, what a group of characters knows is only what can be known by that group of characters in that area in that moment.
Because Middle Earth is BIG, something that Tolkien never, never, never forgets.
Reading The Lord of the Rings is rather like reading about space--one is struck over and over by distance. When Gondor calls for Rohan's aid, it takes several days for the Rohan riders to get to Gondor. It takes Sam and Frodo nearly a week to cross Mordor. Consequently, there is little chance for people in one location to know what is occurring in another location--no satellite imagery, folks!
This is true even for the bad guys, which is a refreshing change from the omniscient bad guy that pervades so much current media. Sure, Sauron is evil. But he is as bound to the confines of his area as much as anyone else.
This is important and one place where Jackson makes a "mistake"*. At the end of The Two Towers movie, Frodo comes face to face with a Nazgul. He doesn't put on the ring, but Nazgul can (semi)sense the ring even when it isn't being worn (the Nazgul have left Mordor to attack Gondor by the time Frodo and Sam enter it). So the Nazgul (and Sauron) would know that the ring had arrived in Gondor. Yet in The Return of the King movie (as in the books), Sauron operates on the assumption that the ring is in Rohan.
Of course, in the movies, the viewer is led to believe that Rohan and Gondor and Mordor are practically next door to each other. In the books, Rohan is far enough away from Mordor that the ring's location pushes up Sauron's time table. He starts his war early because he thinks the ring has not yet arrived in Gondor, and he wants to retrieve it before it arrives there.
So, it's a "mistake." However, at this point, it is important to remember . . .
*MOVIES ARE NOT BOOKS*
One difference between books and films (there are many) is that visuals matter. I know that sounds obvious, but it leads to changes that upset readers. In a book, extra characters can appear and disappear, and nobody cares. In a book, a throw-away line can reference actions in other areas, and again, nobody cares. A book can also cover great distances in a single paragraph.
But do those things in a movie, and the audience will expect the character, reference, or distance to be paid-off. That is, the character, reference, or distance will begin to matter more than it should. Consequently, it must be excised in order to make the movie work.
It is not that movies are "simpler" than books. It is that movies are complex in a different way. Rather than expansion of events and characters (more, more, more), the audience is given depth of moments. In the books, for instance, we know Gimli and Legolas become friends through a series of events; for example, in Lothlorien, Legolas begins to take Gimli with him during his explorations.
The first movie can't shows us all that--Gimli and Legolas's adventures would be their own movie! Instead, it relies on moments: my favorite is when Gimli and Legolas are on the river; when Gimli praises the Lady Galadriel, Legolas smiles to himself. It is infinitely touching and says all we need to know about that relationship.
So I don't fault Jackson for ignoring the distances which Tolkien sticks to so assiduously--it's not like an audience wants to watch Rohan decide to respond to the fires getting lit and then . . .
Ho hum. "Are we there yet?"