There were four Beatles and all of them could play something on the piano. From most to least competent I would rate them as follows: Paul, John, George and Ringo.
Paul McCartney had a head start on the others as his father played and taught him some basics. While in the Beatles he really took it on as a second instrument and it became central to the sound of their later period. Being a technician and serious musician, he endeavoured to get as good at it as possible. His playing was sometimes simple as with "Hey Jude”, rocking as on “Lady Madonna” or slick and sophisticated as on “Martha My Dear”. Whatever he played, (as with all his contributions to Beatles records), it was always highly musical and perfectly appropriate.
When Paul joined the Beatles he reluctantly took up the bass when a bass player was needed and, although he made this work in a way that was extremely impressive in its own right, he was a frustrated guitar player and the piano perhaps allowed him to play a lead instrument that would fit with the band and allow him to scratch that itch. Paul’s basic ballad style is quite simple. He plays the bass note and chord at the same time, then bass note, chord, bass note, chord etc.
John Lennon’s piano playing was more rudimentary that Paul's and was not something he was associated with in the Beatles, but became his signature instrument as a solo artist, particularly because of the single "Imagine" and its accompanying video. John came to love the starkness and minimalism of the sound of the piano and on some album tracks such as "God", used it to have the backing sound as stark as possible. The more intricate piano parts heard on songs like "Jealous Guy" and "Oh My Love" were played by legendary session player Nickie Hopkins. John uses mostly triads (three note major and minor chords in root position and simple rhythms). His right had does a very distinctive pivoting pattern where two fingers cover the upper part of the chord first and then the thumb follows.
We don’t hear a whole lot of George Harrison’s piano playing at all apart from what is in the Get Back documentary when he is composing Something and Old Brown Shoe. In terms of competence, I put him before Ringo as he can apply his knowledge of the guitar onto another instrument. He is well able to knock out a good rhythm and he plays reasonably complex chords such as C7 and F#dim. In the documentary, we see him hammering away with out Old Brown Shoe and it sounds quite convincing.
As for Ringo Starr - his piano style is very much in keeping with his favourite kind of music - country. He does a very simple bass note-chord-bass note-chord rhythm which might be described as a primitive oom-pah and works well for conveying tracks like “Octopus’s Garden”. He does a bit of a rock and roll piano duet with Paul too which sounds convincing enough.
One last interesting fact about The Beatles and the piano. There is one Beatles song where there all played together at the same time on one piano - the big E major chord at the end of “A Day In The Life”!