For context, see the 'Great Ejection' starting in 1662: https://en.wikipedia.org/wiki/Great_Ejection
I can't find any hard numbers on emigration to America because of this, but it definitely restarted a wave of Puritan emigration that had mostly stopped when Cromwell took power.
That lasts to this day, given how puritan America behaves versus Caribbean, South America and Europe.
the thing is alot of Cavaliers escaped to Virginia and during the Restoration Presbyterians and Independents fled to New England.
New England, with its Puritan settlement, had supported the Commonwealth and the Protectorate.
The two groups repeated their conflict two hundred years later in American Civil war
They weren't totally gone, many moved west to pollute a different continent.