If Nexus of Fate is countered by Syncopate does it get exiled or is it upon the controller of Nexus of Fate to choose wich replacement effect applies as stated in CR 616?

Nexus of Fate will be exiled.

While there are two replacement effects that wish to affect where Nexus of Fate is going, the one from Syncopate is what's known as a self-replacement effect–an effect of the spell itself that modifies the effect that spell would otherwise have. As laid out in CR616.1, if one of the multiple effects trying to modify an event and is a self-replacement effect and the other is not, the self-replacement must be applied first.

Once you apply Syncopate's replacement to the event Nexus of Fate's own replacement no longer applies because it's no longer about to be put into a graveyard, so it gets exiled.