Cumberland sausages are an English Christmas dish. What would be a good equivalent in the States?
There really isn't a US equivalent of Cumberland sausages. Sorry.
I'm very hesitant to recommend using American breakfast sausage (eg. Jimmy Dean) as I'm told it's a totally different beast.
You can order Cumberland sausage from specialty companies. Here's one company that seems to be well regarded: http://www.myersofkeswick.com/cumberland-sausages
Or try making your own, http://allrecipes.co.uk/recipe/33513/homemade-cumberland-sausage-ring.aspx