From the House of Commons Library Briefing Paper on the subject:

Before the 1880s questions to the Prime Minister were treated no differently from questions to other Ministers.

Until then questions were asked of ministers, without notice, on days on which ministers were present (usually Mondays, Tuesdays, Thursdays and Fridays), in whatever order Members rose to ask them. Public business could not commence until all questions had been answered. Changes were made in 1881 that affected questions to the Prime Minister, “when as a courtesy to Mr Gladstone, then aged 72, questions to the prime minister were placed last on the day’s list to allow him to come in late”.

Since July 1961, a fixed amount of time each week has been allocated for Questions to the Prime Minister. In his review “The Prime Minister and Parliamentary Questions”, GW Jones outlined those early procedures and explained how questions to the Prime Minister changed before two fifteen minute slots of questions to the Prime Minister, each week, were first established in 1961.

In 1902, 40 minutes was allowed for questions. This increased to 55 minutes in 1906. However, increasing numbers of questions and the asking of more supplementary questions meant that questions to the Prime Minister were rarely reached. In March 1904, they were placed to begin no later than question number 51. But still only about half the questions to the Prime Minister were reached, so from June 1904, they began no later than question number 45. This continued until 1953.

GW Jones then explains why Prime Ministers Questions became restricted to Tuesdays and Thursdays:

From 1953, the ailing Sir Winston Churchill normally answered only on Tuesdays and Thursdays, and what was accepted as a mark of respect to him became a convention continued by Sir Anthony Eden and Mr Macmillan. Questions continued to be put to the Prime Minister for Mondays and Wednesdays and, if number 45 was reached they were usually taken by the Lord President of the Council.

In 1959, the Procedure Committee recommended that Prime Minister’s Questions should be taken during a fixed quarter of an hour on Tuesdays and Thursdays, on other days, questions to the Prime Minister would continue to begin at question number 45.

In February 1960, the House of Commons accepted a proposal from RA Butler, Home Secretary and the Leader of the House of Commons, to begin Prime Minister’s Question at number 40 (rather than number 45), in place of the Procedure Committee’s recommendation. But GW Jones commented that “it was still not early enough, and in July 1961 the House accepted the committee’s recommendation”.

The Speaker introduced the first occasion on which Prime Minister’s Questions were taken at 3.15pm on Tuesday 18 July 1961 in the following way:

The House will have observed that the Order Paper today indicates that the Prime Minister will answer his Questions at 3.15 p.m. The Prime Minister has informed me that he is at the service of the House in this matter and is willing to try this experiment for the remainder of the Session, if that be the wish of the House, as I understand it is.

On 25 October 1961, The Times reported that the arrangement was to be made permanent.

The twice-weekly sessions of Prime Minister’s Questions continued until 1997, however, some changes were made in the intervening period. Philip Norton provided a brief review of these developments:

Initially questions tabled to the Prime Minister covered substantive issues, especially foreign affairs. In the 1970s, the practice developed of asking ‘open’ questions. These did not ask about a specific policy but rather asked the Prime Minister if he would pay an official visit to a particular area or when he last met representatives of a particular body. These were tabled because they were ‘transfer proof’. That is they could not be transferred to another departmental minister. The opportunity to ask a supplementary question allowed the questioner to raise a substantive issue ... Since the latter half of the 1970s, the standard open question has been the ‘engagements’ question, asking the Prime Minister to list his engagements for the day.

Since 1997, Prime Ministers have continued to answer questions but, only once a week, on Wednesdays for half an hour. The change from the twice-weekly format was announced by the Prime Minister, as noted in the first question that he received, from Ian Taylor:

I warmly welcome the Prime Minister to his role of answering questions and I am grateful to him for finding the time in his diary to do so. At some point he might consult the House about these changes.