From this Reddit thread, it looks like Synchronize will work for SOS Pokemon, but it will only take the nature of the first Pokemon 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 ability 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.
Basically, if you have a fainted Pokemon in the first position of your party, but your Pokemon with Synchronize is on the field, Synchronize will be giving SOS Pokemon a 50% chance to have the same nature as the fainted Pokemon, rather than the Pokemon with Synchronize.