This last year I was in Fairfield Co., CT researching the Abbott lines. What I found is that there were two Abbott families living in the Ridgefield area. Your family and my family are the same line, Scottish. The other line is English. My 4th ggf was James Abbott ~1754 to 1840. Family records state that they came to Connecticut via Virginia. The genealogists at FamilyTreeDNA say that we are a 100% match, back 3 or more generations, or in Connecticut. I was not able to located a Nathaniel Abbott from that area, but I now know that they my have lived in New York not Connecticut. They were from a small town west of Ridgefield, Ridgebury, which is on the state line with New York. The Genealogist for the Barnum family, said that his records state that the Abbotts were from Virginia and were Scottish, this matches what my family remembers.
Do you know the location of your Nathaniel Abbott?
James married a Cathrine ?, (Look out there is another James who married Cate Weed, not the same), they had 5 children: James, Azra, Elijah, Catherine and Polly. Only James had children that I can find, he moved west to Fountain Co., IN in 1816. I think your Nataniel may be a 1st cousin.
Regards Ross Abbott