Not true, at least in the obvious case of the Pilgrims of Plymouth. They were perfectly free to practice their religion when they lived in the Netherlands. They moved away because they wanted to create their own strict theocratic colony. The Netherlands, in their eyes, had too much religious freedom.
-running out of room -economics -impending war -wanting to venture into the unknown
https://earthobservatory.nasa.gov/images/91317/holland-first...
The Netherlands do get a point here for tolerance imo.
The Virginia Statue of Religious Freedom originated from one of the Southern colonies, which tended towards more English mainstream religion (Jefferson himself was raised in the Church of England, but considered himself a Deist). It served as a sort of "non-aggression pact" between the religions, since the various sects of the Northern colonies had quite a bit of political power by virtue of concentration of their adherents and isolation from traditional European religious power.
It's a minor quibble, but England does have an established national Church, so it's not entirely secular in the way the US is. 26 CofE Bishops sit in the House of Lords, the Lords Spiritual. The head of state is also the head of the Church.