The first speech to the cardinals by a newly elected pope takes place in the Sistine Chapel which is shown correctly. However, this happens right after the conclave and it is always part of a holy mass. This mass is called Santa Missa pro Ecclesia (Holy Mass for the Church) and is the first Mass celebrated by a new pope together with his brothers cardinals.
Pope Paul VI was the last pope who was coronated and wore a tiara (1963). However, the coronation ritual and the placing of the tiara on the new pope took place during the Papal Inauguration (which by itself was shown incorrectly in episode 3) and not during the Santa Missa pro Ecclesia.