Thank you all for your replies. Here's the "long" version of the story.

We do not have an embassy to India in my country so our visas take a number of weeks to be issued from another country and although we had already applied for visas, because of unforseen delays in procedures we were informed that our visas will not be issued in time for our trip.

Before and after my trip to Bhutan I was intending to spend a few days in India too but I have had to cancel those parts of the trip and I was able to change my flights to go into Bangkok after Bhutan. However the flights from Bangkok to Bhutan were full so I had to stick to the already booked Delhi one. I now have fixed the flight into Delhi so as to have about 9 hours transit time only in Delhi (previously i had about 48hrs) and to spend instead a previously unplanned day transit in Dubai prior to my arrival in Delhi. So the only issue now is to get my luggage transferred. Emirates have confirmed that they will not tag them all the way from Dubai to Paro but I will try anyway though I doubt that its possible. I have booked a transit period in Eaton Smart hotel in the transit area in Delhi and they have confirmed that they will arrange my boarding pass for me. They said that ideally our luggage should be tagged all the way through to Bhutan but if this were to not be the case, they should be able to help with transferring them from one airline to the other.

So I can only hope that this works out now since its pretty much my only option for getting into Bhutan!
The other option was to travel with hand luggage only but there's a 5kg limit on Druk Air so that's tough.