I think this is doable. For a POD, I suggest that the attempts in the 1560s to colonize what became northern Florida and South Carolina are not made and the funding and colonists directed to what became New York harbor instead. An earlier date doesn't really work because France didn't really start trying until decades after Verrazzano's voyage.
We will assume that they construct their main fort in lower Manhattan like the Dutch and not, say, on the west bank of the Hudson so this version of New York city develops sort of like ours.
They beat the Dutch to what we call New York, and since the French are already moving into Quebec, the next logical step is to establish a chain for forts between Manhattan and Montreal Island, not just St. Jean and Ticonderoga but they will establish forts at least on the sites of Fort George, Albany, and probably also in the Hudson HIghlands to establish a continuous chain between Manhattan and Quebec. This is done instead of pushing into the Great Lakes and Mississippi, so what is lost are the French forts at Niagara and Detroit. THey might be established later, but a less aggressive French push into the Great Lakes because the Hudson corridor is prioritized probably butterflies away the pushes to the Ohio and Mississippi rivers (Pittsburgh and St. Louis), though Biloxi and New Orleans are probably still on.
The low numbers of French colonists are not, and were not IOTL, a problem in maintaining a colonial empire in the seventeenth century. It only became a problem really after 1750 due mainly to the British allowing unchecked European immigration into their colonies. We might get more French colonists anyway, even with them bstill eing picky about who gets to settle there, if the Hudson Valley is available to settlement. And being able to link up to Quebec can only help the Hudson Valley settlements.
This will butterfly away the fall of New Sweden. For reasons I am not familiar with, the Dutch got into a war with Sweden at some point. Sweden was consistently a French ally during this period. And Sweden never went to war with England either, again unlike the Dutch, so the English probably also leave the Delaware Valley alone as well.
The question is what England and the United Provinces do with these new facts on the ground. Do they still try to grab the French and Swedish colonies in the Mid Atlantic or do they divert their efforts elsewhere? This is a good question. The Dutch were always at war with Franch in the seventeenth century, but England or at least its kings were friendly to France until 1688, enemies afterwards. I think the English don't move against the French and Dutch colonies under the Stuarts and the colony for the Duke of York is found by moving forward the colonization of Georgia into the seventeenth century. But the Dutch may make a grab for Manhattan. The problems here are the French are in a better position to fight back than the Dutch were against the English, and the WIC is always prioritizing the Caribbean and South America, they after all preferred to go with Guinea to New York IOTL, so they probably try to grab more Caribbean islands and forget about the beavers.
The grants to Penn, Carteret, and Berkeley, what developed into Pennsylvania, New Jersey, and Delaware, were carved out of the grant allocated to James Stuart, Duke of York. ITTL, the Duke of York is granted Georgia. Maybe its enlarged by transferring South Carolina into this sphere, but the English could try to grab St. Augustine (though tye were not at war with Spain during this period) or the Bahamas get more settled. But the existence of Philadelphia is butteflied away.
And apart from mainland Nova Scotia/ Acadia, the British swallowed up French North America in one go in 1758-60 (they took Louisbourg earlier but traded it back), so we are looking at a French Hudson Valley and a Swedish Delaware Valley through 1750. But the French are not moving in towards the Ohio ITTL so the British/ Virginia move into that area. As far as the American colonists are concerned, there is no need for the "French and Indian War". Compared to our timeline, there is much more English settlement in what became western Pennsylvania, Georgia, and maybe Kentucky as well and none in what became Eastern Pennsylvania, New Jersey, New York, and Delaware. Instead, Eastern Pennsylvania, Delaware, and southern New Jersey are Swedish, and New York and maybe northern New Jersey are French. The Swedes are fine with German Protestants settling there area so this still happens. The Hudson Valley is less settled than IOTL and probably also the Delaware Valley, but the total number of northern Europeans in mainland North America is actually slightly higher since there are more Swedes and French.
The Seven Years War and the French and Indian War plays out differently. A lot depends on whether French armed Indians are still raiding New England ITTL. If they are, the British and their colonists still move against TTL Quebec. If not, they may well leave it alone. If they leave it alone, no contracts to the colonists that get withdrawn in the austerity budgets of the 1760s and 1770s and the American War of Independence and possibly the French Revolution go away in the form we know them. And if the British due take over the French and Swedish colonies later on some excuse, there is little to no solidarity between New England and its grievances and the now more populous English colonies in the South (no connection) or any colonies conquered in the Middle Atlantic (not enough English speakers).
