There are seperate counters for those arriving with and without visas, and the queue for the latter is much longer.  You can clear the immigration sooner if you have a visa but that obviously does not expedite the baggage retrieval process.  You either wait longer at the immigration, or at the baggage claim.  My suggestion is that you complete the application and have the photographs ready prior to your departure.  Also complete the disembarkation card during your flight.  That way, you can immediately join the queue while others are still filling out their application forms, and your baggage will probably be on the carousel right around the time you have cleared the immigration.  The queues can be pretty long if several flights have landed around the same time, and having the application form already completed can save you a lot of time.

I strongly advise against using two passports for the same trip.  If you plan to visit India while you are in Nepal (and return to Nepal before leaving for your country or elsewhere), then it makes sense to carry a Nepali passport FOR YOUR TRIP TO INDIA otherwise do not use it.  Use only your foreign passport for travel to and from Nepal.