How have you guys been handling migrations in mongo?
That said, those circumstances are hopefully pretty rare. We generally write scripts that bridge between the two formats. One to copy/transform and one to cleanup after. Run the transform, test and verify, then move to cleanup.
It's a huge pain, but if you keep on top of if --especially the cleanup-- then it gets easier. It tends to get really hard once you get a lot of cruft. You end up with the code vs data dichotomy. So keeping on top of that really helps longer term.
Your existing server would continue, you have a new endpoint to test with updated application code, deployment can then be a standard blue-green approach.