I'm a NC attorney, but this is not intended to be specific legal advice but just some general information.
(1) As other's have mentioned, if you have an original of your Uncle's will, your cousin who was named as executor has the ability to file the will through the probate of the County in which your Uncle lived (if that's the same as the county in which his house is located, then all the better).
(2) If the neighbor files first, the law requires that the neighbor notify all heirs under the laws (that would include nieces and nephews whose parents are deceased -- otherwise the heirs are the surviving siblings, and the children of any sibling that is deceased). This proceeding is a "will caveat", i.e. a challenge to the sufficiency of a will.
(3) The will caveat can be based on (a) a later will that by its terms cancelled the earlier will; (b)lack of testamentary capacity at the time the will was signed; and/or (c) undue influence over the testator of the will.
(4) In the same vein, if the neighbor files a will first, you and your cousins have standing to challenge (or caveat) the will the neighbor attempts to probate. There are a number of things that amount to lack of testamentary capacity, but the problem is that the decedent only needs to have a lucid moment when he is signing his will. The fact that he disinherited his natural heirs is some evidence. There are also a number of things that could amount to undue influence, one of which is if the neighbor denied access to your uncle. (I think you may have an uphill battle there).
(5) Under some circumstances, the attorney's fees for the caveator (the person contesting the will) can be paid from the funds available in the estate. (In other words, unless there is more in the estate, you might each be entitled to recover attorney's fees in this dispute from the estate -- which might amount to 30K to 50K each -- thus depleting the thing that you are fighting over).
(6) From a non-legal perspective, if no one had contact with you uncle, why do you think you and your cousins are entitled to share in his estate? Just a question, no judgment. But something to think about.