I'm curious why Nepal and Bhutan remained nominally separate from the British Indian Empire. They were both under British influence, so what distinguished them from the other native monarchies like Kashmir and Hyderabad?
(I'll preemptively link this thread; the responses there only seem to address the fact that Nepal and Bhutan, like the other native states, weren't under direct British rule.)
I'm currently on mobile so is difficult to link to my previous explanation of the Bhutanese reasons for not joining the Indian Empire. You can either access my top posts to find it or wait a few hours for me to link to it. Another just explains the Nepali situation in equal detail.
But I'll explain the broad reasons here before you go into regional specifics with that post:
Politics. You basically bit the nail on the head with "Nepal and Bhutan, like the other native states, weren't under direct British rule." Is kind of the same reason that Brazil and Venezuela ended up technically united during the period of United Iberia but we're separated when Portugal and Spain divorced. To the kings of Nepal and Bhutan, they were equivalent to the new heads of government of India and the Pakistans and weren't simply princely states like Hyderabad and Kashmir as you mentioned.
Geography. British control over India was focused on A. Trade routes to China, B. The Great Game with Russia, and C. The production of precious precious tea.
Simply put: the Opium Wars made trade through Tibet to China obsolete, unnecessary, and overall unprofitable. So even though the Himalayas weren't directly profitable, the were worth influencing to keep the Russians out but again, not profitable enough to rule directly. Especially since the wars that Britain did engaged in (the Gurkha Wars in Nepal, and the Duar War in Bhutan)could successfully invade but in no way occupy kingdoms at the top of the world, making local role with official "advisory" roles in the British Empire the preferred option. But not without taking the profitable low land of both kingdoms in the ensuing treaties. Why else? To plant tea (again, making direct trade with China less important).
I'll link the detailed explanations in a few hours but I hope that's a satisfactory introduction to the topic.
Nepal had invaded Bengal in the early 19th century, and had been defeated by the HEIC/Brits. The consequent treaty, amongst other things, gave the HEIC some Nepalese territory, put a British Resident in Nepal, established friendly terms between the signatories, but Nepal retained its independence. I would suggest that, as Nepal was on the borders of British India, had become friendly, and provided an effective barrier to Chinese expansion, there was no need for any further action in Indianising the Kingdom.
Bhutan was in a similar situation, except that Bhutan did become, effectively an Indian princely State, rather like Kashmir and Hyderabad. However, although both Bhutan and Hyderabad declared themselves to be independent sovereign states after Indian Independence, India didn't invade and conquer Bhutan as they did Hyderabad and Goa. Hence Bhutan became an independent state.
Not much of value for the British Empire in the Kathmandu valley. And was difficult to send a strong military force with malarial jungle and then mountains in the way, and Napoleon in Egypt around 1800 when the approach was first tried. And Brits did not think Russians would try to get to India that way either.