As far as asserting Executive Privilege, it basically just means that there will be one more legal layer to dig through before the conflict can be resolved.
For all practical purposes, it will not be possible to resolve this issue before the November elections unless the parties suddenly come to an agreement after having failed to do so (or really even attempt it) several previous times.
At this point, I almost think the Democrats welcome the contempt vote. They are going to play it as politically motivated (as evidenced by comments from Rep. Elijah Cummings, Sen. Pat Leahy, and now the President) and hope that it distracts people from some of the other issues where they are doing poorly in public opinion.
Based on past examples, they can easily delay this past the November elections, so if they were the only source of the documents, then it makes sense to sit on them. The wildcard is that there is clearly one or more people who are handing over sensitive documents to the Oversight Committee. If those people hand over something earth-shattering (or have already done so), then this strategy could blow up in the Administration's face very quickly.
Barring something like that, I would guess you'll see a Oversight Committee vote that falls on party lines. After that, they will attempt to negotiate/stall again and you'll see a floor vote in the House with a few Democratic crossovers. After that vote, the House will recommend it to DOJ for prosecution. DOJ will investigate as long as they can and then decline prosecution. The House will file a civil contempt motion and the whole thing will go to court, where it will easily drag out past November.