Nepali citizenship for son/daughter

All I can do is to give you strictly personal advice, not legal advice. Right or wrong, my best hunch is that the Citizenship Certificate for your kid (the legal paper denoting citizenship to be obtained after a child turns 16) is in my view pretty much all in the hands of your Village Development Committee or Municipality concerned and for those people to decide "yeah" or "nay" based on whatever information the VDC will ask you to provide. 
Have you talked to them to date? If not, I would suggest that you do exactly that and direct your query to them. Those administration officials should be able to answer any other questions you may have and guide you accordingly.  Who knows, when the time comes, they may ask for evidence of a land owning certificate, proof of payment of house tax, etc. and any other prerequisite at that future time.
In theory when one's offspring turns 16, you can usually apply for citizenship if the mother or father (either one) is able to sign the official document according to the Nepal Citizenship Act of 2006. So, yes, you have every right to make the claim for your child. In practice, however, it should be expected that nothing in this country be a foregone conclusion, witness the hundreds of thousands of "stateless" people here.
Best of luck anyway in your case - and that's sincere.

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You child can only have one citizenship as it stands now. So he is an American by birth and you will be able to go there easier through him. I'd wait and see if Nepal begins to allow duel citizenship. They've been talking about this for a long time. Hopefully, it will happen.

Most nepalis I know would give their right arm to go abroad, and to be able to live and be free to work there, holder of a golden US passport/citizenship to boot!!