This is a great question. I haven’t had any experience in this situation, but I think there might be some great advantages to integrate all at once. This way you can really evaluate best practices, might find gap areas that would need to be addressed later if you delayed integration, and you get buy-in quicker from both organizations. Change is uncomfortable, so if both organizations are moving their cheese at the same time – it feels more equal in the feelings and work being done. Will it be more difficult to tackle both at once? Absolutely. Could it result in an even more successful integration? Possibly.
To Mateusz’s comment above, each case would needed to be independently evaluated.