So just what was the King's objection to the emigration of the Saybrook nobility?
It's not relevant to your direction, I suppose, but I'm quite intrigued by the concept. The colonies developed their own First Families even absent the real deal. Having genuine aristocrats fooling about would have interesting consequences.
Religious and economic reasons, effectively.
On the first, Charles was determined to impose religious conformity on his Kingdoms. While there's an obvious temptation to let the worst radicals bugger off somewhere where nobody cares, Archbishop Laud, took the view that there was no excuse for dissent and that opponents to his religious settlement should be forced to knuckle down and accept it rather than go into exile. You get the general impression that Laud was far more inflexible on this than the King, but the Archbishop was an influential sort.
For most of the period we're talking here, the main theatre for this struggle was a battle over the legality of the Massachusetts Bay Colony. Ferndinando Gorges claimed his land patent, which had been granted in 1622, already covered the areas settled by the Bay Colony, protested the issuance of the Massachusetts Royal Charter in 1632, and basically spent most of the 1630s trying to get the colony shut down, egged on by Laud. He might have been successful, given time, but the Civil War intervened and obviously Parliament took a more sympathetic view. This is touched on to a certain extent later on, but only tangentally- I didn't want to get too bogged down in it.
Then there's the economic motive. The King had inherited a huge deficit and was always desperate for money, particularly as the 1630s went on and the Bishops' wars started. This led to him reviving a whole bunch of obsolete taxes such as Distraint of Knighthood (This was a retrospective tax whereby if you weren't at his coronation, you paid a fine.) and most infamously, Ship Money. All this fed into the growing mood of anger against the King that eventually led to the Civil War.
From an economic perspective, having the odd churchman or burgher emigrate wasn't a massive problem. But when large landowners such as Saye and Sele, Brooke, or Hampden, who have already led serious resistance to Ship Money, start preparing to liquidate their assets and leave the country, the Crown is going to see the move in terms of tax avoidance as much as anything else. And as the King is desperate for cash, that's the last thing he needs.
So, you've Charles I. You've tolerated handfuls of unimportant subjects disappearing to found colonies for a couple of decades, but suddenly realise that they could be joined by some major contributors to the exchequer. That prospect's bad enough, especially when considering that they might convince their friends to join them. On top of that, not only are they insubordinate enemies of the Crown, but also borderline heretics who will use their exile to practice their error-ridden doctrines away from the sensible oversight of the church. And while doing this, they'll have a safe haven from which to spread their unacceptable views back to England.
So, the solution is simple- ban them from emigrating to New England. If they're that desperate to move, let them go to Ireland; at least there they're still paying tax and doing the Crown some good in holding the country down.
I should say, btw, that while we're not going to see a formal New English aristocracy ITTL in the sense of a transplated one from across the Atlantic, Saybrook does have a hereditary upper house with a term for life, and will continue to do so. This will have interesting effects in the long term.