There does happen to be a ferry between India and Sri Lanka. If you are planning a trip through South Asia without using a aeroplane, you can travel from Sri Lanka to India over the sea and then overland to Kathmandu.
(If anyone reading this question is looking to find a way into India without using flights, then a cruise ship or cargo ship to Sri Lanka or India might be an option.)
There is a ferry between India and Sri Lanka, between the city of Nagapattinam in Tamil Nadu of India (very close to the railway station) and Kankesanturai in the Northern Province of Sri Lanka.
If you can get to Sri Lanka on a cruise or cargo ship, then it will be possible to get to India. I think they allow bicycles on the ferry but I cannot confirm anything as the service only started very recently (they tend to be very generous about luggage but contact them in advance).
The main problem is this option is of course getting to Sri Lanka, however it might be useful for anyone planning a bike trip through South Asia.
I believe it is possible to travel to Nepal from India as it has an open porous border (unlike the sea between India and Sri Lanka), but all other borders into Nepal are tough to get through.
If I were you then I'd take a flight to Sri Lanka, and travel overland to Kankesanturai, then by ferry to India and finally overland to Kathmandu in Nepal.
You Can Travel to Sri Lanka (or India) by Cruise Ship or Cargo Ship
Sri Lanka is a hub for cruise ships that often travel onwards to the Maldives, so you might be able to secure a spot on a cruise ship just to Sri Lanka if you were willing to spend some money. Some cruise ships also travel to India.
Sri Lanka also happens to have the main transshipment ports for South Asia, so there are plenty of cargo ships which you can pay to travel on, and I assume that there are numerous cargo ships that travel to various parts of India as well (though less frequently).