Train to Nepal from Kolkata
The shortest/fastest way would be taking a bus/train to siligudi.. then take a bus to KTM. takes about a day and a half or shorter if u can catch the buses on time..
Nepal has got no railway of its own as most of the area is hilly. You will have to travel to siliguri via New Jalpaiguri by Indian Railways and from siliguri you enter the Nepal border to Kakarvitta on jeep. From Kakarvitta onwards you will get plenty of buses plying to any part of nepal. There is no other choice. Alternatively you could try the air route.
Kolkata to Jalpaiguri/Silguri (train)
Silguri to Kakarvitta (bus/jeep)
Kakarvitta to Kathmandu (Bus).