Well, I discuss this scenario here.
The interesting questions to me are:
1. Would Eddard keep an oath sworn to a religion he doesn’t believe in? Because if not, then Eddard living probably means he sends a letter putting Cersei’s dirty laundry out on the street and proclaiming Stannis to be the true king, which might very well mean Stannis starts the war with the North, the Riverlands, part of the Stormlands, and possibly the Vale or parts of it, on his side.
2. If he still keeps to the oath, Robb Stark’s not going to turn back around as long as the Lannisters hold his sisters and menace his kin in the Riverlands and have so badly insulted his family. On the other hand, it’s less likely that he goes full King in the North.
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