September - November is the best time to visit South Asia
During this time, the heat has subsided enough to be able to bear the temperatures. At the same time, it is dry which makes travelling convenient.
Most places will heat up during the day but can be navigated comfortably in the mornings and evenings. However, certain parts of the Bhutan and Nepal become completely inaccessible, and clouds can obscure mountain peaks. The west coast of Sri Lanka will be hit by monsoons during this period so it is good time to explore the east coast.
It is mostly sunny and dry throughout the day making travelling easier. However, the temperatures can get very high in certain areas.
In Nepal and Bhutan, the snow makes a lot of the trekking routes inaccessible. Most places in India have a pleasant weather during this time, except for the Himalayas which can be a bit too cold. It the best time of the year to visit the west and south coast of Sri Lanka as monsoon approaches the east coast.