Dhanaulti at first.. you can explore mussoorie and give some time to it as this place will fall on your route if you are going from Dehradun side. You may get snowfall by late December or in Jan-Feb months.
Yes, you would get snow in Mussoorie in month of January if you want any update for Mussoorie feel free to revert me back
I have been to Mussoorie for like more then 10 time I can tell you each and every thing about Mussoorie