So, say Robb is never named King in the North and the Starks and Tullys aren’t trying to establish independence from the IT. What does Stannis do in this instance? Does he go for an alliance or does he still see them as enemies?

Stannis was considering an alliance in OTL even though Robb had been named King in the North, so yeah, he’d be trying to recruit them onto his side right off the bat. 

And given Robb’s sentiments, had the decision still been open when Stannis’ letter arrived, I think he would have gone for Stannis, especially with Catelyn there to link Stannis’ letter to Ned’s investigations.