You don't need a far-reaching POD to prevent the Separatists from sailing.
The big question is who colonizes New England instead of them. It's a fairly worthless piece of land to a colonizing power. You can't grow any cash crops and it's got a horrible climate. The most likely options are an English penal colony to threaten Quebec/fish the Grand Banks, a French penal colony to provide a bulwark/base against the Dutch, or the Dutch slowly colonize it due to its proximity to New Netherland.
A less likely option is that some small European power takes it for prestige. Sweden or Denmark are the two options I can think of off the top of my head.
Even less likely, some other persecuted European religious minority could choose to move there in the wake of the Thirty Years' War. Maybe the Anabaptists, or the Quakers are more severely persecuted in England, driving them to New England?
These options aren't mutually exclusive; the Dutch could easily claim it, then invite in an interested religious minority to keep the French and English out.