They have a bit of a nutty-grainy flavor. Biscuit/Victory malt and a little chocolate malt help bring this out. Some of my favorite recipes also include Special B, and some include brown sugar or molasses.
Nut browns usually have low-to-moderate hopping with English hops, where American browns are hopped more aggressively with American varieties.
That said, there is no BJCP definition of the style, and they're considered either Southern- or Northern English brown ales. Northern browns are a little hoppier than southern. So if your beer tastes a little nutty/grainy to you, you can call it a nut brown if you want.
One of my favorite recipes is from Norther Brewer. Basically, it's 6lbs LME; 4oz each of chocolate, Special B, Bisciut and Special Roast; 1oz fuggle. This makes a beer almost identical to one of my all time faves, Wachusett Nut Brown Ale.