It seems that Neilson was a 1905 graduate of the U.S. Naval Academy who resigned from the service in 1908 to study art. He returned to the navy as a Lieutenant (equivalent to army captain rank) when the United States entered the Great War and served as an aide to Vice Admiral William Sims who commanded U.S. naval operations in Europe (the latter point from this source): clearly Neilson had connections. The second link also mentions that he was "Member American Artists Professional League. N.A.; Clubs: Salmagundi, Century. Home and Studio: 131 E. 66th St. New York City 21, New York." That address was not and is not in a shabby neighborhood. But then, he was married to the daughter of a Pittsburgh steel maker.
The first link notes: "Neilson studied with William Merritt Chase and at the Art Students League with George Bridgman and George Bellows. He continued his art education in Paris, studying at the Académie Julian, Ecole des Beaux-Arts, Académie Colarossi, and the Academie Grande Chaumière." This surely took place mostly before the war began in 1914 and when many American artists returned home.
Now for speculation about connections with Richard E. Miller (1875-1943). Miller's Wikipedia entry is here. It mentions that Miller spent much of his time from perhaps 1900 to 1914 in France, spending some summers with the colony of American artists in Giverny, nearby where Claude Monet lived. Neilson and Miller might well have met either there or in Paris. In 1917 Miller moved to arty Provincetown at the northern end of Cape Cod, even during the 1920s only a day's journey from New York City where Neilson was based.
Now consider the images below.
here. Not all Miller paintings seem to be dated, but his one is almost surely from his Provincetown days. Note his signature at the lower left (click to enlarge).
I should add that Neilson painted a few other Impressionist-style paintings of women that can be found by Googling on him and then selecting Images. From the looks of these, they might have been done in Giverny before the war.
What to make of this?
Almost certainly Neilson was experimenting with Miller's style, perhaps because he was, or was about to become a painting instructor and wanted to re-familiarize himself with Impressionist portraiture. Furthermore, he surely knew Miller.
From this, I can think of two alternatives. The first is that Neilson went to Provincetown and worked on his painting during the time Miller was painting the two images of his shown above.
A second, possibly more likely explanation is that Neilson visited Miller and semi-copied elements from both while Miller provided some coaching. A variation on this is that Neilson saw the paintings together elsewhere while doing his version -- though I consider this possibility unlikely.
Please comment if you have more solid information about this matter.