For non-extreme hikers that want to play it safe, and are visiting India during September and early October - is it better to begin in the north-east (Sikim etc.) or north-west (Himachal-Pradesh etc.) weather-wise? Where is it warmer during the early winter?
September is one of the rainiest months. Consider North India to follow the climate from the Northern Hemisphere.
I suggest you start at the mid-north and work your way up. I.e - Gujarat/Rajastan/Delhi and then head north. Note, rainfall decreases as temperature dips into winter - and this happens significantly during September. Combine this with the fact that you are heading to higher altitudes and colder climate, you might see some very clear, cold but drier weather in October in the north.
I was travelling with some older family members and had to do it the latter way as they were finding it difficult with the cold weather up north.
Both Sikkim and Himachal Pradesh are going to be cold by virtue of altitude. You haven't exactly mentioned any specific places so I can't comment on which place will have colder weather, but considering typical 'places to visit' Sikkim is likelier to be the colder place (again, altitude)
Late September / early October is the end of monsoon season. So you can expect some rain in both places. You'll need to check weather reports as to what the actual conditions are. In case of roads being blocked due to heavy rains / local landslides blocking mountain roads, you're likelier to find more alternatives in Himachal Pradesh as transport links there are better (closer to populous centres in northern plains, etc. so it sees higher number of visitors).