Wednesday is busy because the papal audience usually takes place that morning, in the square or in the Paul VI Hall, and Sunday because the Pope prays the Angelus at noon. Visitors report that on audience mornings the basilica line hardly moves until the audience is over, so on those days come at opening or choose another day. The basilica is open from 7:00 a.m. to 8:00 p.m. all year, and the late afternoon is generally reported to be easier than midday.
No ticket removes the police check. The basilica's own paid reservation guarantees a time slot, includes an audio guide and uses what the basilica calls a dedicated entrance route, and the basilica recommends it for avoiding long queues, but its pages do not say that booked visitors are exempt from screening. Pilgrims and those coming to pray may use the Prayer Route, which begins on Via di Porta Angelica, is marked with yellow signs, is staffed twelve hours a day and enters through the Vatican Grottoes; again, nothing says it bypasses security. Anyone selling a skip-the-line ticket for the basilica itself is selling a tour or a reservation service, since entry is free.
Good to know before you go: there is no luggage storage or cloakroom inside the basilica, and it advises against bringing excessively large bags, so leave suitcases at your lodging and carry a small day bag. Dress is checked on the way in: shoulders covered, long trousers for men, skirts below the knee or trousers for women. In summer bring water and a hat for the wait.