The amount of time it takes to learn a song on piano depends on the complexity of the song and whether your trying to read from sheet music.  

For example it would take you 1 year to learn Fur Elise from sheet music but if you learned it from a teacher by watching and doing then it could be 2 months. 

Another variable is if you are trying to learn to play the melody on the piano vs singing the song or accompanying others. If you play one note in the left hand and the chord in the right, you can learn songs super fast versus playing the melody on the piano which can take months to perfect.