Countable atomic model without a countable saturated model

Take a countable structure with countably many independent unary predicates, every finite predicate pattern realized infinitely often, and a constant naming every element. Its complete theory has an atomic model: tuples of named elements have types isolated by equations to their names. But every infinite binary predicate pattern, together with inequalities to every named element, is consistent. There are uncountably many such one-types, so no countable omega-saturated model exists. Thus existence of a countable atomic model does not imply existence of a countable saturated one.