Sorry for posting again so soon.
As far as the Cause for His Holiness John Paul II is concerned, the preliminary parts should go through easily, swiftly, and without any trouble. This is determining whether the candidate lived heroic virtues (as defined by Catholicism). The other qualification is martyrdom: since the candidate in question was not martyred, the Cause will concern itself with his heroic virtues. I do not think there is any doubt whatsoever that the candidate did, indeed, live and personify heroic virtues.
From the second website linked above, it is written that the candidate’s bishop would submit the Cause. Since the current candidate was the bishop of Rome, the current bishop of Rome would lead his cause, which makes for an interesting situation. I believe this is resolved by the vicar of the Roman diocese (Camillo Cardinal Ruini), who, if I remember correctly, fills in for the Holy Father as bishop of Rome when he is occupied as Supreme Pontiff.
The miracles will take some time, although perhaps not too much time. From the moment he died, Catholics have been praying for the intercession of His Holiness Pope John Paul II. With so much praying, there’s bound to be some miracles happening.
Of course, being the Supreme Pontiff, the Pope can at any time dispense with the formalities, as it were, and declare him a saint. It would be within his prerogative and I don’t think he would be opposed. Certain Vatican people might be quite upset: I don’t think the Curia likes for the Pope - whoever he may be - to go around rather than through them.