The colloquial British-English expression padre’s hour designates a weekly hour of religious instruction provided by chaplains to British-Army units.

The noun padre (literally father) is colloquially used to designate and address a male chaplain in the armed forces.
—Cf. also the jocular Australian-English phrase gone on the padre’s bike.

David George Coulter evoked the creation of the padre’s hours in The Church of Scotland Army Chaplains in the Second World War, a thesis submitted for the degree of Doctor of Philosophy, published by the University of Edinburgh in 1997 [pages 139-140]:

In the spring of 1942, an Anglican chaplain, the Rev J. J. A. Hodgins, organised the first formal scheme of religious education, or chaplaincy training periods, known as the “Padre’s Hours”. This was done with the support of Major General “Boy” Browning * who was then commanding the Airborne Division.
The basis of this scheme was that each week a group of soldiers should be given time from their training programme to meet with the padre, who would present a short talk on a religious or moral topic and then open it up for discussion.

* The British-Army officer Frederick Arthur Montague ‘Boy’ Browning (1896-1965) married the British novelist, biographer and playwright Daphne du Maurier (1907-1989) in 1932.

The earliest occurrences of the expression padre’s hour that I have found are as follows, in chronological order:

1-: From an account of a meeting of the Streatham A.T.C. Executive Committee, which took place on Tuesday 29^th September 1942, published in The Streatham News (London, England) of Friday 2^nd October 1942 [No. 2,754, page 1, column 8]—A.T.C. stands for Air Training Corps:

It was also reported that arrangements were being made to show films of R.A.F. interest (such as “Target for To-night”) to the boys, and the padre (Rev. O. K. de Berry) had arranged a “Padre’s Hour,” for each Friday, starting to-night. It will be run on social lines.

2-: From The Stirling Observer (Stirling, Stirlingshire, Scotland) of Tuesday 12^th January 1943 [No. 41, page 4, column 5]—C.F. stands for Chaplain to the Forces:

The revelation that about half the personnel of the British Army to-day knew nothing about the principles of Christianity was made by Captain the Rev. W. P. Watkins, C.F., when he spoke on “A Padre’s Problems” to Stirling Rotary Club members at their weekly luncheon in the Golden Lion Hotel on Friday. In that connection, he mentioned that from this week onwards, soldiers were to be given an hour’s religious instruction, to be known as “The Padre’s Hour.” Not more than a hundred men would be present at the classes daily.

3-: From an item of news reprinted from the Church Times (London, England), published in several U.S. newspapers on Friday 26^th February 1943—for example in the Oklahoma City Star (Oklahoma City, Oklahoma, USA) [Vol. 25, No. 14, page 4, column 7]:

The Right Rev. Dr. C. W. G. Taylor is the first Moderator of the Church of Scotland ever to visit the British Fleet at an English base. Dr. Taylor addressing the Church Office Bearers Union in Glasgow with disarming modesty conveyed this news. He added:
“I spent 10 days with the Fleet south of the border and was gratified at the reception I got. I have also been to many army and R. A. F. units. I was also recently the guest of an extremely young general of a famous Scottish division, and he asked me to help him launch a “padre’s hour.” As a result every officer and man once a week gets a chance to discuss freely his religious problems.”

4-: From an account of a meeting of the Lincoln Diocesan Mothers’ Union, which took place on Thursday 25^th February 1943, published in the Lincolnshire Echo (Lincoln, Lincolnshire, England) of Saturday 27^th February 1943 [No. 14,533, page 3, column 5]—the following is from a speech made during that meeting by Mrs. Fisher, chairwoman of the Churches Committee for Work Among Women in the Forces:

Describing the formation of the committee, Mrs. Fisher said that during the late months of 1939 it had been realised by some women of the churches that some such organisation would be urgently required. These women got together to discuss what they could do by way of helping the chaplains to meet the great extra burden which had been put upon them by the entry, in large numbers, of women into the Services.
[…]
“The chaplains did welcome the help,” went on Mrs. Fisher, “and we have received from the beginning the wholehearted support of the three chief chaplains, but the women and the girls apparently did not want to be educated. Their lives are tremendously full with Army education of one kind and another, and they did not want, in those early days, to sit down to definite religious instruction, although they did appreciate the talks which were arranged for them.
“Just lately, however, a new factor has come into being, and this is helping our work very much. Our workers are sometimes asked to help the padres in their ‘padre’s hour,’ which they have every week for religious instruction.”

5-: From the Penrith Observer (Penrith, Cumberland, England) of Tuesday 2^nd March 1943 [No. 4,250, page 3, column 2]—A.B.C.A. stands for Army Bureau of Current Affairs, the name of an organisation promoting discussion among soldiers about current events, citizenship and post-war reconstruction during the Second World War:

An interesting article, specially contributed, appeared in yesterday’s “Daily Despatch,” relative to the recent innovation of “Padre’s Hours.” The author states:
“Our soldiers are acquiring a new outlook on religion.
“I put that statement before you in its bold and stiff words because I am anxious not to exaggerate the tremendous significance which underlies it. Our soldiers are not an army of psalm-singing Cromwellians. They have the lovable habit of trying to hide their deeper feelings and decrying their virtues. But I have seen in the week just past clear evidence that they are becoming more actively religious.
“It is the bare truth to say that a new spirit is beginning to stir within the army. No more, no less, it has not the semblance of the birth of a wave of revivalism: it is to be sensed more than seen. But I shall be surprised if it does not grow stronger and spread steadily.
“I have just come back from a northern market town where I saw hundreds of soldiers at religious services far different from the familiar type of Church Parades. You know that the “Padre’s Hours” were introduced into the army recently. Taken in working time like A.B.C.A. discussions they bring soldiers and padre together every week for full and frank discussions. They have become popular. The soldiers have been glad to have the chance to discuss religion. So in the market town it was decided to give them more opportunities and a Religion and Life Week was arranged.”

6-: From Time: The Weekly Newsmagazine (New York City, New York, USA) of Monday 8^th March 1943 [Vol. 41, No. 10, page 48, column 3]:

Religion
Padre’s Hour

All British Army units in the United Kingdom now have an obligatory, weekly religious instruction period. Called the Padre’s Hour, it was planned by Army chaplains at the behest of handsome, youngish Major General Frederick Arthur Montague (“Boy”) Browning, commander of Britain’s airborne troops, husband of Novelist Daphne du Maurier.
The Padre’s Hour sometimes includes a whole company, but the ideal is the informal (they can smoke if they want), clublike meeting of a platoon, 20-25 men. Chaplains usually start with a 15-minute address on any topic they wish, then hold a 45-minute question-&-answer session. The Hour is apt to be quite as instructive for the padres as for their men. Many chaplains have been joggled by such questions as: “How can you justify the Church’s ownership of slum property?” Many have been startled by posers like “Why is God always represented as male?” Some 75% of the questions are social-economic, about 15% are purely religious.
Chaplains in general feel that the Hour is raising the religious interest of the troops. They cite the fact that many men ask for baptism, confirmation, some talk about entering the ministry after the war. But by & large the men’s attitude tends toward the cagey. Many of them preface their questions apologetically: “Without disrespect to the Church. . . .” Says the Ministry of Information, with bland British understatement: “There is extremely little real hostility to the Church. The commonest attitude might be accurately described as disappointment or impatience.”