Quagmire, Here is what I think your cousin should do:

Get the Visa to India.  Nepali passport holder don't need visa to travel India but since your cousin with asylum status won't have a Nepali passport it is safe to get Indian Visa.  It cost $73 and can avoid him a lot of confusion and problem.  Below is the link for Indian Visa.  Also beware that unlike Nepal, India doesn’t have any immigration office at the airport to grant him Visa.

https://indiavisa.travisaoutsourcing.com/consular-fees

Second, I strongly suggest not flying to Nepal as with flight there is a risk of record.  Take a bus or Train. 

I'm not a Lawyer but this is what one of my friend who was on the same boat as your cousin did.