A totally uneducated guess is that they will automatically handle the transition. In experiments with humans wearing glasses to vertically flip their visuals they start to see the flipped view as normal after a few days. No doubt birds would be the same. There’s also the evidence that birds have not gone extinct

If the images in the article are accurate, the birds don't really see the polarity of the field, merely the alignment, so they don't really see "north", "east", "south" and "west", merely "latitudinal" or "longitudinal".

They would probably combine with sunlight to get the real orientation – an approach which would continue to work with a flipped magnetic field.

lots of subpopulations of many migratory birds dont infact migrate, or migrate very short distances. May play a role in their recovery.

I'm not an expert in this domain, but my guess is that a flip of the eaths magnetic field is not instantaneous, but it takes over a few bird generations.

I think it would be hard to evolve to protect against something that only happens every half million years. We're probably going to need to observe the behavioral changes the next time it happens to have any clue.