The island of Nevis is a good place to come to get away from mass-produced fast food, because it is one Caribbean island which has no Burger King and no McDonalds.

Your best choice for cheap eats on Nevis is to try one of the roadside vendors, such as the Yellow Bus in Charlestown or the Water Department on Pump Road (great ribs and a good local spot on Friday nights).  There is Mansa's at Cades Bay, which sells freshly-barbequed ribs and chicken, and has ice cream available in a freezer, as well as being a very good farmstand with all kinds of really excellent fruit and vegetables, not to mention lovely juices made fresh at the farm.  Several bakeries and local shops offer homemade breads, sandwiches, rotis and saltfish cakes. You could maybe pick up something to snack on at the beach (but please don't leave litter).

The smaller local restaurants such as Chevy's Calypso Bar & Grill located on Pinney's Beach serve great local food, international food and seafood, and are much less expensive that the hotels.  You can also try dining at Rumours, Pemo's, Shirley's, and Callabahoo.  Several of these also have lovely places to sit while you eat.