We have an oil painting with the signature R. Neilson, purchased…

Sorry, I thought that was the spelling I saw in the posted image...it may be XXXXX XXXXX Rodgers Neilson (1881 - 1964) who has a very similar style...art is difficult to evaluate from just images.

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Known for portraits and still life painting, New York City painter Raymond
Neilson had well-known portrait subjects such as Admiral William Halsey; Dr.
Francis Blake, Dean of the Yale Medical School; and World War II hero Brigadeer
General Frederick Castle. He also did a portrait of Hayley Lever, fellow
Academician at the National Academy of Design*. Many of the portraits were done
for Civic Organizations and presented at ceremonies commemorating the subject of
the portrait.
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Neilson won many awards for his painting including exhibition prizes at the Paris Salon*, the Panama-Pacific Exposition* of 1915, the National Academy of Design, and the Salmagundi Club*. Other exhibition venues included the New York World's Fair of 1939, Carnegie Institute International*, and Grand Central Art Galleries.*
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Neilson was born in New York City and graduated from the United States Naval Academy in 1905. However, in 1908 he gave up a Navy career to follow his art, but he was reinstated during World War I and served overseas as an aide to Admiral William Sims. Then he studied at the Art Students League* in New York, in Paris, and at the Boston Museum of Fine Arts.

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Here is an example of his style: (this one is titled "Moroccans on Horseback")

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I strongly encourage you to take it physically to a gallery for confirmation, as many artists have very similar styles and again, without actually being able to see it in person, I can only rely on guides and the images posted therein. If it is a work by him, the normal range for that artist is $1,200 to $1,500 for a work like that. If you can send me a close up of the actual signature, I will try to find an example to compare it with. Best, Will