There are many ways to answer that question. It depends on what your favorite songs are! ~smile~
If you want to be a concert pianist and play original versions of advanced classical pieces, that is going to take additional time and effort.
If you want to just be able to listen to yourself play some familiar songs and sound pretty good, then you can expect to start playing songs fairly quickly.
Much of it depends on you, your individual learning ability, your availability to practice and your level of commitment.
Oops, did I say commitment? Yes, I did.. it takes commitment to learn how to play the piano. Don’t let that scare you however… if you can find 10 minutes, 3-4 times a week to practice, you can learn how to play the piano and be well on your way to playing your favorite songs in a few short weeks.