As a math problem, this is a case of a random walk ("walking" N cards at a step from your opponent's deck to your deck, where N is usually one, but can be 4 or 7 or whatever depending on ties.) It is often framed as the "gamblers ruin" problem. Ignoring the issue of ties and treating each throw of cards as an independent trial, the average number of turns is 26*26 = 676 turns.
Factoring in ties would reduce it quite a bit based on N -- allowing for bigger steps occasionally. The average number of turns when the step size is 5 (i.e., every turn is a war that is resolved on its first tiebreak) is equivalent to playing with 6 cards each (26/5, rounded up). And 6*6 = 36 turns.
So then it's just a matter of figuring out the probability of big steps. Ties can be expected about 3/51 of the time (when you play a card, there are 3 cards that can tie it out of the 51 other cards). 48/51 = step size 1, 3/51 * 46/49 = step size 5, 3/51 * 3/49 * 44/47 = step size 9, and so on. This makes an average step size of about 1.25. A game played with all steps equal to 1.25 would be expected to last about 21*21 = 441 turns (26/1.25 = 21 cards each equiv). And then reduce that some more to factor in the chance for sudden death when one of the ties occurs when a player is low on cards. (Sorry I don't have the details on how to do that exactly.)
EDIT: I should also emphasize that this approach assumes the trials are independent trials. That's basically the difference between dice and cards. Each throw of a die is independent. But each draw is not -- it depends on what else has been drawn / what else remains. And in the case of War, winning a trial in general improves your odds on the Kth following draw (when you draw that card again). It's a very mild (IMO) runaway leader effect (mitigated by also receiving the losing card, but I still think the net is a gain). And it will multiply the advantage, if any, bestowed to one player over the other in the initial deal. That is, any slight deviation toward the higher-numbered cards in the initial deal will be amplified each successive trip through the deck.
Simple Random Walk: http://www2.math.uu.se/~sea/kurser/stokprocmn1/slumpvandring_eng.pdf
First Passage of a One-Dimensional Random Walker