According to Bir Shumsher's diary, semi-democratic elections between independents (as in no political party affiliation allowed, but other than that, more or less democratic), and around 20% of the people being allowed the voting franchise would vote every 4 years on the National Assembly. The Nepalese, Tibetan and Bhutanese legislatures would remain in place, though they would turn from national legislatures to devolved legislatures. And yes, the national assembly was supposed to have superseding powers over the constituent legislatures, and the militaries were to be combined to form one military. The proposed National Assembly took heavy inspiration from the Canadian Parliament and its federation powers. The otl plan was basically Canada's federation system copy pasted in Tibetan and Nepali.
Huh, that is interesting. Well, Canada's Confederation worked because the US was big and scary and the Fenians had recently been invading, so everyone was willing to put their own independence aside for the sake of collective security. Plus Britain was helping to push it. For this union would China form enough of a big scary neighbor for the three to do the same?
As to effects, it would be ironic, but seems quite possible, that the first major military outing for this new federation would be in support of China, if worry about China is what brought them together. AIUI the Araniko Path was a yak track at the time. Though it may get upgraded if it is the main connection between Kathmandu and Lhasa. Even still, it is an extremely hazardous route. Connection with Bhutan seems like it would be equally difficult. If it is seen as an alternative to the Burma Road, then it may see allied investment in improving the route, possibly even adding a rail connection if that is feasible. If the Army can be concentrated then it does seem logical to support the KMT against the Japanese, though I would not rule out forces still being deployed alongside the British in Burma.
No matter who wins the Chinese Civil War, they are likely going to cause problems over Tibet. AIUI the KMT was actually more aggressive about this in the short term. The Communists may take a slower approach. But, particularly as British international influence wanes and India's focus gets locked on Pakistan, and vise versa, then there will likely be communist insurgents working in Tibet.
Seems to me the Federation has a few options when it comes to the Cold War. They could try to remain unaligned and remain compliant enough to China that they won't bother going after them. In this case an alliance of some kind with India would be very desirable. India might be a bit unreliable as a protector though. Either way, it would be a narrow tightrope to walk. This likely means a degree of communist influence in the government. Alternatively they could try and court Britain as a protector initially and then switch that to the US as British influence wanes and the US takes more interest in the situation. This likely means some harsh crackdowns on communist insurgency, greater tension on the border, and US bases in Himalayan territory. Considering its proximity to both Russia and China, this could lead to some tense moments.