Over many centuries there were disputes over exactly what role pope, king (or in Germany the Holy Roman Emperor) or local clergy should play in the choice of bishops. These disputes were known as the Investiture Controversies.
Whatever the legal position, which was rarely very settled, the approval of both the king and the pope was required in practice. In the early thirteenth century the Pope appointed, or purported to appoint, an archbishop without the King's approval, whereupon the King refused to allow that person into the country. The result was the Interdict, with church services largely banned throughout England. The King was John, the Pope was Innocent III and the Roman nominee as archbishop was Stephen Langton. The matter was complicated because the chapter had appealed to the Pope.
Certainly by Henry VIII's time, the role of cathedral chapters, that is representives of local clergy (or clergy who occupied local positions but may have been usually absent) was only a nominal responsibility to elect the new bishop. They were not permitted to proceed with an election without permission from the King, who generally only gave them permission to elect a specific person, his nominee. However, a bishop could still only be consecrated with permission from the Pope.
So in this way the King effectively chose bishops but the pope could effectively veto that choice by not granting permission for consecration. Rome did not appoint English bishops. The chapter could, in theory, refuse to elect anyone.
Cranmer was appointed by the King in the early Autumn of 1532. The papal bull approving his consecration was received in late March 1533 and he was consecrated on March 30th 1533. It may seem strange that Rome would approve a known supporter of Royal Supremacy, and someone prominent in the campaign for the annulment of Henry's marriage with Catherine of Aragon. Matters were not seen by everyone in such stark terms as they were by Bishop Fisher of Rochester.
Later in 1533, the law was changed to remove the need for papal approval. To this day Church of England bishops are elected by chapters by the command of the monarch who specifies who they may elect. If they refuse then the monarch (in practice the Prime Minister) can make the appointment anyway. However, the newly appointed bishop cannot be formally consecrated and confirmed without the involvement of other bishops and a mandate from the monarch.
The monarch acts on the advice of the Prime Minister who is advised by a committee including diocesan representatives. It is perhaps one of the more bizarre aspects of the British constitution, which however seems to work tolerably well.
A curiosity easily overlooked is that the Bishop of Rome, the Pope, is still elected solely by representatives of the clergy in the Diocese of Rome. How can this be? Because every cardinal is appointed to a titular church in the diocese of Rome. That too may seem slightly bizarre, but it too seems to work.