Who becomes Thane of Cawdor after Macbeth becomes king?

Nobody succeeds Macbeth as Thane of Cawdor within the context of the play. As Banquo says to him, in act 3, scene 1, "thou hast it now: king, Cawdor, Glamis, all." Macbeth, having become Thane of Glamis first, can simultaneously hold this title alongside that of Thane of Cawdor; having become king, he simply maintains all three positions, meaning that he is able to wield power over Glamis and Cawdor specifically, which in theory would enable him to be a more powerful king. In practice, this does not work out for Macbeth, of course. We would expect the titles of Thane of Cawdor and Thane of Glamis to be passed down from Macbeth to his sons, which Macbeth does not get the chance to do.
In a modern monarchy, it would be very unlikely for a king to maintain any previous lesser titles—for example, King George VI of Great Britain was previously the Duke of York, a title which he renounced upon becoming king. But it remains perfectly possible for a Prince to simultaneously hold a number of other titles: Prince William of Great Britain is also Duke of Cambridge, Earl of Strathearn, and Baron Carrickfergus. Because Macbeth takes place in a time when the king and thanes of Scotland held significant personal and political power, it would not make sense for him to renounce his thaneships, as that would give other people the opportunity to build their own powers. Had he continued as king, he could have awarded this thaneship to a loyal follower, as Duncan awarded it to Macbeth.