Keeping the scope limited to India:
It is worthy of noting that the climate in western Himalayan region is pretty much defined and heavily influenced by what happens in southwest during it's monsoon season (from June to September) and by that in western systems (from November to March).
The names marked with an asterisk (*) are states in India I'll be referring to.
If you observe, there are 6 states in India that have Himalayan ranges.
State of Kashmir is not heavily influenced by south-western flows. This region sees a mild rainfall during monsoon season, but experiences a heavier snowfall in winters. Pre-monsoon (April to early-June) are pleasant. You are not expected to have substantial snow-level up to an altitude of 5000m. Snow from the previous winters may be already in the melting cycle, causing streams to have higher water levels and force during the peak hours of day - 11:30 am to 2:30 pm. Beyond 5000m expect patches/sections of snow on almost all north faces of mountains. The higher you go, snow levels are more prominent. Post monsoon - all the moraines will be pain to cross, but rejuvenated grasslands will be a treat to the eyes.
Ladakh (Union territory of Ladakh) has an extreme weather. There is no monsoon season as such here. It receives very minimal rainfall. This is because it falls under a rain-shadow region being surrounded by high ranges from southern parts (sub-sections) of Himalayas and Karakorams in the north. Expect dry weather throughout the day with temperature reaching up to 32-35 Celsius, high winds around evening and very cold nights. Most of the region is cold desert so it poses a different challenges in mountaineering. This region has a lot of technical peaks above 6000m and high altitude passes. Most altitude gain would be dry moraines and expect a lot of mixed climbing above 5500m. Pre-monsoon and post-monsoon is not a valid discussion for Ladakh.
The state of Himachal Pradesh and Uttarakhand are considered paradise for mountaineers in India. I have observed that the average rainfall during the monsoon is slightly higher in state of Uttarakhand than Himachal Pradesh. Monsoon season is considered to be around months of June to September. These states see a heavy snow falls by the end of month of November and high-altitude peaks are difficult/impossible until the snow melts, which is in late March. It is noteworthy to know that most roads and high-altitude passes are opened during the month of April. That makes reaching certain camps easier. In a nutshell, you can start expeditions in pre-monsoon, around month of April/May and expect better conditions for mountaineering. A lot of streams with fresh cold water melting from some or the other glacier. Expect patches of snow above 4000m. The northern faces of these sub-ranges are much much colder and receives a lot of snow.
I can't answer for state of Sikkim or other ranges in the north-east states of India.
In my opinion, weather in Southern sections of Tibet will be much like Ladakh.