They were in service until at least the early 2200's. Most proably being replaced by the Deadalus class vessels.
Enterprise was being decommissioned in 2161, according to the Holo-program in "These are the voyages..."
Daedalus class ships were decommissioned in 2196 per TNG "Power Play". At the end of Enterprise, Trip mentions the new Warp 7 ship being constructed, presumably this is a reference to the Daedalus class. With the joining of Vulcan, Andorian, Tellarite, and Terran technology, the NX class would quickly become obsolete.
Just a note, there's an NX class, and there's NX registered vessels. Experimental vessels (generally the first of each class, the USS Prometheus and USS Defiant for example) are labelled NX- instead of NCC-