The title '"Supreme Governor" of the Church of England is rather misleading. It dates back to the time when Henry VIII determined that the Pope should have no authority in England. That decision was primarily political rather than theological. Henry denied the power of the Pope to refuse Henry an annulment of his first marriage, which had only been permitted by an earlier papal dispensation. Henry wanted a male heir, and Catherine of Aragon had only one child who later became Queen Mary following the death of her half-brother Edward VI. It should be noted that under Mary the Church in England was reconciled with the Pope and the final break only came during the reign of her half-sister Elizabeth, when the causes were even more clearly political than they had been during Henry's reign.
At that time the "divine right of kings" gave monarchs virtually unfettered authority. This authority was presumed to extend to the Church as well as secular matters. Its effect was reduced over the century and a half following Henry's repudiation of papal authority to the extent that James II was ousted in 1688, largely because of his overt membership of the Roman Catholic Church. Even now nobody who is a Roman Catholic, or married to someone who is, can succeed to the throne of the United Kingdom.
The United kingdom now has a constitutional monarchy where the monarch is advised by the Prime Minister on how to exercise their authority, and is effectively bound to follow that advice. In the relatively recent (past few decades) past there has been a move to reduce the power of the political power to direct the affairs of the Church of England. The selection of names to be considered for appointment as bishops now rests with a Church-appointed committee, though the Prime Minister does retain the right to choose between the two names submitted. Having made that choice the preferred candidate (assuming they are willing to accept the nomination and pass the medical) is nominated by the monarch and duly elected.
Church legislation receives close scrutiny in the Church's General Synod before being submitted for Parliamentary approval, which does not include the power to amend what is placed before Parliament. Once approved by Parliament it receives the Royal Assent. The power of the monarch, and of the Prime Minister, has been significantly attenuated over the years though UK coinage still carries the initials FD. These initials stand for "Fideii Defensor" or "Defender of (the) Faith" a title bestowed on Henry VIII for a paper he wrote defending the Catholic concept of seven sacraments against the two recognized by the reformers