Nepal's Highlands are blisteringly cold during late December and much of the country will be inaccessible. The valley will be chilly at night but nice during the day. Keep in mind that most cheap hotels do not have heat in the valley and remember to dress warm for when it goes down to freezing at night. You wont have to worry about rain though. I believe there is a direct flight from Shanghai. But flying there will be expensive.

To go nepal from China, currently there are two ways. one is from China southern. There is direct flight from Guangzhou to Kathmandu. And the second is from China air, from Beijing to chengdu to Kathmandu. You do not need permit to Lhasa if you go by Chian Air. Price depends on ticket agent. Two way ticket may cost around 5000 RMB.
In Nepal, do not spend time just in Kathmandu. IF you go out of valley, you will see natural beauty. Bungy Jumping, rafting, trekking, elephant riding, national park, historical place, temples, etc are very good in Nepal. Pokhara is one beautiful city.

Well to be sure a month is not even enough time to see even a small portion of India.

I still think the civil war in Nepal has left a lot of open wounds and similar experiences can be had in India. Sure Mumbai might have some attacks but most of India is a very safe place.

Depending on your nationality you can get a 10 year multiple entry tourist visa to India for relatively cheap making repeat travel easy.