In the 19th century, Sweden owned the Caribbean island of St. Barthelemy for a short while before it became French again.
Assuming Sweden kept it, how would its governance be today? Would it be integrated like France?
I'm pretty sure that Sweden would have gotten rid of it the same way Denmark got rid of the Danish West Indies: Selling to either America, Britain or France. In the long run, and with the Carribean colonies becoming a drain on more than an asset to small nations' economies, there is no real reason to hang on to them except some country-ball-worthy "We can into great power!!!!"-nonsense that no one is buying.
So I don't think it's possible, but let's for a moment assume that it IS.
I don't think Stockholm is going to do much against the express will of the people living there. I don't know much about how it was governed in the brief period it WAS Swedish, but I could imagine something like the deal the Faroe Islands and Greenland has with Denmark: Devolved parliament deal with most local matters, Stockholm takes care of defense, foreign policy, etc.
Though unless some serious Swedification has been set in motion early ( = seeing it as more than a nice place for plantations, but as an actual part of Sweden) to create cultural and historical bonds between Sweden and the colony, it's eventually going to come to the question of: "What do we have in common with these people, and what do they give us that we can't get elsewhere, and closer?"
So, I think it's exceedingly unlikely in the first place, but assuming it happens, it's likely to be something akin to the Faroe Islands and Greenland, though with a lot more "What are we even doing in this union??"-questions.
