How does Synchronize interact with SOS Battles?

From this Reddit thread, it looks like Synchronize will work for SOS Pokemon, but it will only take the nature of the first Pokémon in your party into account:

There is a bug in Synchronize effect with SOS calls : Synchronize works on allies if the Synchronizer is on the field, but with the nature of the first party Pokémon. The Synchronizer can even lose Synchronize with Roleplay/Skill Swap, it will still work. It also works if the first party Pokémon is KO.

This information is also supported here:

If you want to synchronize the nature of the spawning allies in SOS battles you have to fight with you synchronizer in the first party spot. It isn't enough to have it fainted on the first spot of your party. Additionally if in your first party spot is a fainted Pokémon and you're fighting with your synchronizer in the second slot the synchronized nature is the fainted Pokémon's not the synchronizer’s.

It appears that this was confirmed during the datamine of the Sun/Moon demo, the results of which were discussed here. The specific post that involves Synchronize is this one:

Yup, the bug SOS Call chains and Synchronized is now confirmed. When you send a Pokémon with Synchronize on the field and chain for allies, the allies will be affected by Synchronize. However, the nature that they will have 50% of the time is the nature of the first party Pokémon.

Basically, if you have a fainted Pokémon in the first position of your party, but your Pokémon with Synchronize is on the field, Synchronize will be giving SOS Pokémon a 50% chance to have the same nature as the fainted Pokémon, rather than the Pokémon with Synchronize.

From my own experience, every Pokémon that is encountered through an SOS call will be affected by Synchronize.

Which Pokémon is out in battle at the time the new Pokémon is called makes no difference, the only thing that matters is that the Synchronize Pokémon was in the lead before starting the battle.