The only type of set that can possibly be uncountable is the type that has greater cardinality than the set of natural numbers. Every other set, finite and infinite (if it has cardinality equal to the set of natural numbers) is countable.
So yes, if countability requires one to one correspondence to the set of natural numbers, and all finite sets are countable, then they do indeed have one to one correspondence with the set of natural numbers.
I am aware that some people define countability as only applying to infinite sets. In other words, by this definition, the only countable sets are infinite. The use of these competing definitions might be where some of the confusion in this thread is coming from.