The short answer is “By the chapter. But actually, by the monarch. But actually, by the prime minister.” As for the long answer, it gets a bit complicated.

Background

In theory, English bishops were elected by the free choice of the chapter, the group of clergy attached to the cathedral of the diocese. Thus we read in the very first article of Magna Carta that “the English Church shall be free, and shall have its rights undiminished, and its liberties unimpaired…. we granted and confirmed by charter the freedom of the Church's elections….” But while this may have been genuinely been the case in the early middle ages, episcopal elections were increasingly overshadowed by royal influence. While the members of the chapter would still meet to vote for the new bishop, they would usually be careful to only vote for the candidate whom they knew the king desired. In the thirteenth century the system was further complicated by the addition of another player: the Pope, who claimed the right to issue “provisions” for vacant bishoprics. By the end of the fourteenth century it was necessary to have both papal and royal approval for all new bishops, and some negotiation was needed to find candidates acceptable to both sides, but it seems that the king generally got his way—Clement VI allegedly declared his willingness to appoint an ass to a bishopric if requested! All changed at the Reformation, of course, and an act of 1534 granted the monarch exclusive power over episcopal appointments:

at every avoidance of every archbishopric or bishopric within this realm, or in any other the king's dominions, the king our sovereign lord, his heirs and successors, may grant to the prior and convent, or the dean and chapter of the cathedral churches or monasteries where the see of such archbishopric or bishopric shall happen to be void, a licence under the great seal, as of old time has been accustomed, to proceed to election of an archbishop or bishop of the see so being void, with a letter missive, containing the name of the person which they shall elect and choose: by virtue of which licence the said dean and chapter, or prior and convent, to whom any such licence and letters missive shall be directed, shall with all speed and celerity in due form elect and choose the said person named in the said letters missive, to the dignity and office of the archbishopric or bishopric so being void, and none other.

And this remained the law of the land up to the mid twentieth century: The chapter would receive the Congé d'élire, or permission to conduct an election, and in the same breath it would receive the “letter missive” informing it precisely who was to win that election.

How it Worked

Here’s an example showing the actual text of these two documents, as it happened in Ely in 1864. First came the Congé d'élire:

Victoria, by the grace of God of the United Kingdom of Great Britain and Ireland Queen, Defender of the Faith, to our trusty and well-beloved the Dean and Chapter of our Cathedral Church of Ely, greeting. Supplication having been humbly made to us on your part that, whereas the aforesaid church is now void and destitute of the solace of a pastor by the death of the Right Reverend Father in God, Doctor Thomas Turton, late bishop thereof: We would be graciously pleased to grant you our fundatorial leave and license to elect another bishop and pastor of the said see: We being favourably inclined to your prayers in this behalf, have thought fit, by virtue of these presents, to grant you such leave and license, requiring and commanding you, by the faith and allegiance by which you stand bound to us, that you elect such a person for your bishop and pastor as may be devoted to God and useful and faithful to us and our kingdom: in witness whereof we have caused these our letters to be made patent. Witness ourself at Westminster, the eleventh day of February, in the twenty-seventh year of our reign.

Then the letter missive:

Victoria R.—Trusty and well-beloved, we greet you well: Whereas the Bishopric of Ely is at this present void by the death of the Right Reverend Father in God, Doctor Thomas Turton, late bishop thereof: We let you weet [know] for certain considerations, us at this present moving, we, of our princely disposition and zeal, being desirous to prefer unto the same see a person meet thereunto, and considering the virtue, learning, wisdom, gravity, and other good gifts wherewith our trusty and well-beloved Edward Harold Browne, Clerk, Bachelor in Divinity, is endued, we have been pleased to name and recommend him unto you to be elected and chosen into the said Bishopric of Ely: Wherefore we require you, upon receipt hereof, to proceed to your election according to the laws and statutes of this our realm, and our Congé d'élire herewith sent you, and the same election so made, to certify us under your common seal. Given at our Court at Saint James’s, the eleventh day of February, 1864, in the twenty-seventh year of our reign.

And very shortly thereafter, on February 19th, the chapter of Ely returned to Her Majesty a report that the same Edward Harold Browne had been duly elected.

People of the time did not fail to recognize the rather absurd nature of an “election” in which the outcome is decided beforehand by the government. As Emerson put it in his English Traits (1857),

They go into the cathedral, chant and pray, and beseech the Holy Ghost to assist them in their choice; and, after these invocations, invariably find that the dictates of the Holy Ghost agree with the recommendations of the Queen.

Some inside the Church were concerned as well. In 1867 a Cornish clergyman wrote a pamphlet condemning it as “a solemn, profane, and wicked farce.” Other proposals were more conciliatory; an 1863 petition on the “right of free election” pointed out that no change in the law would actually be required in order to end the “farce,” as the Act of 1534 says that the Crown “may” rather than “shall” issue letters missive to the electors. But by and large, there does not appear to have been too much discontent in the church. In 1870 an official report of the Convocation of Canterbury recommended that cathedral chapters be allowed in extreme cases to reject the royal nominee, but the Crown’s right in general to select bishops should not be tampered with. As it turned out, it wasn’t until 1976 that the establishment of the Crown Nominations Commission gave the Church of England an official role in choosing its own leaders.