I have no idea how long it takes the average person to pick up piano, but all people are different when it comes to playing instruments and playing different instruments. I started teaching myself to play piano a few months ago. It took me a couple hours to get "The Scientist" by Coldplay to the point where I could sing and play at the same time. A few days later I went for Piano Man, that one took a good chunk of time.
The thing about any musical instrument is that teaching yourself a song is
not the same as being proficient with the instrument. You could know 1000 songs and be able to play them perfectly, but memorization and actual skill are very different.
Just practice.

Make sure you don't only play songs though, because that won't necessarily make you better. Songs are all fine and good, but do finger exercises and learn a little about the theory too. (Assuming you really want to improve)