November until March is great, but be prepared for cold mornings in winter, December, January and February in particular, you will have cold starts. So bring your Christmas gloves, scarves, hipflask and hot water bottle. The afternoons are transformed to suitably sultry, however.
Most national parks in India are closed during the monsoon season, during July, August and September, so check with your tour operator. March to June in most tiger watching regions are extremely hot. Although this is a good time for viewings as wildlife comes out of the forests to find water. But with temperatures in the mid 40s centigrade, you need to be well prepared.
April and May are less hot in Nepal, further north, however. And the landscape turns luscious too. So a win win.
In Siberia, when tracking tigers, winter is best, as a lot of other wildlife has gone quiet, and so the tracks stand out. Late winter and, in particular, February is the best time, as tigers are at their most active in looking for food.
December to February are winter months in Bhutan, but sightings are magnificent in the snow, and also the monsoon season has ended.
The majority of people on tiger safaris in India, by far, are Indians themselves. So, Indian public holidays and festivals are times to avoid, as they book up far in advance or the park may even be closed. Big ones include 26 January for Republic Day and Diwali which falls October or November.
Just after the monsoon in October the vegetation is still quite high, making sightings more difficult but the parks are definitely prettier and less burned out than in the hot safari season.