Reading this I've learnt a bit but one question: Why did so many Nepalis decide to migrate to Sikkim and Bhutan in the first place? Is this some pan-Indian trend?
While I can't tell you the specific reason for Nepalise the 19th century is full of these kinds of migration, it's usual because of the spread of new agricultural developments and crops spread to large degree by Europeans and which some people get first (like Nepalise lowlanders) resulting in a population boom among them, and existence of European dominance over native state allow a greater movement of the local population between the difference states. Colonial areas are full of these stories, even in Europe we see things like this, like with the Swedish-Finnish colonisation of Sami areas, the settlement of Germans in Russia etc.