You won't find flights to Bhutan from anywhere. Trust me, there's only two available flights and they're both irregular and unpredictable along with being difficult to book.
The best way is take a flight to Kolkata and then a train from there to Hasimara which is about 17kms away from the Bhutan border. From there take a cab or auto to reach Bhutan. Hopefully I was helpful!
@Dixit along with the replies given by other travellers, please remember that if you're going in high season, it's better to book flight tickets as soon as the booking window opens because the tickets get sold out quick. We couldn't go to Bhutan in Oct'18 because when we checked for flight tickets in Jan'18, it was already sold out!
I came back from. Bhutan last month. You won't find any flights on normal booking sites.Bhutan have two airlines. I suggest you check druk airlines online and go to there site to book tickets. But it will surely turn out to be much costly. And yes none of Bhutan flights starts from Mumbai. Either you have to get to Delhi or kolkata. The budget option is to take a flight to Bagdogra and then go by road which has turned out to be cheap for me