I never played scales or arpeggios, i never learnt the names of the treble notes, i just know the position of each treble key on the clef. I know the name of the bass buttons but not their position on the bass clef. I can read music but I cannot play by ear and can only play 2 tunes from memory.
I got a skype teacher about 3 years ago and he got me practicing 2 octaves of c scale and 2 octaves of c major arpeggio and some other simple excercises and gradually increased the difficulty of tunes I was playing and i have noticed continual improvement so that now I can nearly play Olive Blossoms and Nola, my 2 tunes.
I practice for about an hour every day, more would be better.
Hope this helps