From just looking at the two pictures, without historical context, I would say that

• the photograph came first
• the painting traces the photo not very well

The "painting" has too much of both detail and simplification where would not expect it. See the grainy ground around the shadow of the closest soldier? Some stones appear to be too sharp, but some fine structures are missing.

Also, the painting changes the "church"-like building on the right, just in front of the rightmost soldier at the front. It's altered in shape and in size.

It could be another question: why exactly this building was altered. Everything else seems to be very directly replicated.

The state of the art in photography at the time would make such a photo impossible in any case.

All photographs had to be staged, the grain is too fine etc. Many commercial painters could render a painting of that quality very quickly from sketches that would take only a few minutes. There may have been several sketch artists working at the same time for newspapers to print. After their painters created an oil picture from sketches. This would explain why the photograph has minor difference with this particular painting. It was a photograph of a different painting.

Before photography was perfected many many artists made a living in exactly this manner.

The sepoy rebellion was a big deal in Britain. The army was astounded that British trained troops would behave in such a manner. That is why their revenge was so terrible.

Radio Yerevan reports that: "In principle, yes, the photo is absolutely real. Except, that it isn't a real photo, its name isn't 'England's Revenge in India', and the propaganda book never claimed any of that". (See end of post for a quick resolution)

The picture in the question has a wrong description at the German Propaganda Archive, it is also not a photo originally

The educational site German Propaganda Archive (GPA) hosted by Calvin College makes crucial mistakes in presenting the picture and describing it. In modern terms: the metadata is mixed up. As this remains utterly wrong as on the Alamy, or also wrong in "Visual Nazi Propaganda From Hagur’s Private Collection since 1998", © March 2013 – Skull Press Ebook Publications Ghent, Belgium (Non Commercial), (PDF)

Why? The version in the GPA had evidently a switch between pictures and their captions on one double page, as perhaps the original owner seems to have confused the correct placement/position in his picture collection book on one double page.

The original two pictures as published in 1941

If we look at the original Nazi book, the mix up on the American GPA site is evident, as the original does not claim to show a photograph for "England's Revenge in India"; it has a different title and doesn't mention a magazine as source, but indeed just the Russian painter originally:

click for larger images
(src for the last two pictures)

The picture on the left is titled "England's Revenge on India" "(Englands Rache an Indien)" and it is sourced as 'published by the Picture Post, 3 Juni 1939'.
On GPA the caption for this picture was used in error to describe the "photo or painting" in question.

What's going on here with three possibilities in the question to choose from?

Just: none of the above from the three possibilities presented by OP.

The 'photo' on the right page 77 is the one in question and in the Nazi book titled almost correctly as "Execution of rebellious Sepoys" (Hinrichtung aufständischer Sepoys) and accurately described as being a painting by Russian artist Wassili Wereschtschagin. The English title Suppression of the Indian Revolt by the English is rendered in German as "Hinrichtung von Sepoys (1857)".

So, in the original source we have only a reproduction of a painting for publication, correctly described and attributed. No claim that it is a "photo", and no claim to the effect of having found this in an English magazine.

The picture captioned "England's Revenge on India"

The actual photo titled in the book "England's Revenge on India", on the left of that double page, is 'real' to be 'real' as well:

Source of that image:
According to Sean Willc**k, Postdoctoral Fellow, Paul Mellon Centre for Studies in British Art: "Colonial Photography in British India", September 21, 2015:

Image: Felice Beato, Two Sepoys of the 31st Native Infantry Who Were Hanged at Lucknow, albumen print from collodion-on-glass negative, June 1858. The J. Paul Getty Museum, Los Angeles.
Digital image courtesy of the Getty’s Open Content Program.

Source attribution for "England's Revenge on India"

From where the Cigarette manufacturers got it for their Nazi propaganda book is unknown at the moment. But the claim that the Picture Post indeed did publish this picture with the hangings in mid-1939 is not disproven either.

Two studies that also analyse the book Raubstaat do not question the authenticity of any of the pictures used in that pamphlet. But both note the method of how they are used: by screwing with the context in which they appear and – ironically – giving sometimes tenuous descriptions or explanations.

… show very little on their own, but rely entirely on the accompanying text for the damning narrative. The readers again have to complete these connections, and in so doing collude in the politicisation of the hobby.
— Victoria Stiles: "Empire and national character: British imperialism in books from the 'Third Reich'", PhD thesis, University of Nottingham, 2015. (PDF)

… recurring terms and themes are gruesomeness, sadism, torture, robbery, victims and loot, revenge, hypocrisy, unscrupulousness, greed and perfidious Albion. ‘Cant’ makes several appearances, introduced and explained with schoolmasterly care as an untranslatable term specific to English political hypocrisy: “das die politische Scheinheiligkeit des Engländers bezeichnet, ist daher nicht in anderen Sprachen übersetzbar.” Illustrations, particularly by Gillray and Hogarth, are carefully selected from English or French sources. These included Punch and French satirical magazines (that often also had Germany in their sights): L’Assiette au Beurre (1901–36) and Le Carivari (1832–1937).
— Catherine Mason: "Raubstaat England. The Long Reach of Colonial Resentment", Angermion, Volume 12: Issue 1, 2020. doi

The currently accepted answer was misled into searching for a painting in the Picture Post archive despite the original source never claiming to have pulled it from there. The search was thus necessarily fruitless since it looked for the wrong picture. The propaganda booklet claimed that the picture of the hangings was from the Picture Post, not the execution-by-cannon.

Whether that photo was also published in that magazine or not, we might find out in time. — Update: And as we can clearly see on that other post here on SE: it was published in the magazine of that name on the date as referenced, just like the Nazi booklet claimed.

Propaganda doesn't work if you just lie as you like all the time.

Q Is the photograph “England’s Revenge in India” real, staged, or fake?

The photo with that title is real. But the photo with that title isn't pictured in the question. The picture from the question is a reproduction of a painting and properly declared as such in the question generating propaganda pamphlet.
No mystery there, just an error in the German Propaganda Archive.

From a picture usage viewpoint the Nazi pamphlet uses on these two specific pages just the very same pictures as the current Wikipedia page for Indian Rebellion of 1857, and almost in the same context, directly together:

The whole propaganda book is in an online version here, or as a really horrible audio-book but with page-by-page illustration here.

In the original nazi-book we see again the correct placement of pictures and captions:

Englands Rache an Indien Dieses Photo von 1857 wurde 1939 von der englischen Zeitschrift“ Picture Post“ veröffentlicht. Es zeigt die Methoden, mit denen die englischen Behörden nach der Niederwerfung des Sepoy-Aufstandes vorgingen. „Die Hinrichtungen von Eingeborenen geschahen ganz summarisch und wahllos“, berichtet ein englischer Augenzeuge.
(England's revenge on India. This photo from 1857 was published in 1939 by the English magazine Picture Post. It shows the methods used by the English authorities after the defeat of the Sepoy uprising. "The executions of the natives were summary and indiscriminate", according to an English eyewitness.)

Hinrichtung aufständischer Sepoys. Für die Vollstreckung der massenhaften Todesurteile an den aufständischen indischen Soldaten erfanden die englischen Offiziere eine neue Hinrichtungsart: die Verurteilten wurden vor Kanonen gebunden und so „in die Luft geblasen“. Gemälde des russischen Malers Wassilij Wereschtschagin.
(Execution of rebel sepoys. For the execution of the mass death sentences on the insurgent Indian soldiers, the English officers invented a new method of execution: the condemned were tied in front of cannons and thus "blown up". Painting by the Russian painter Vasily Vereschagin.)

How did this error occur?

There are basically two possibilities:

The author of GPA, Bytwerk, might have just confused two pictures and their captions. But even more likely is that due to the nature of this book:
Raubstaat England started out as an exhibition in Munich. The pictures shown there were then distributed as cigarette cards and the 'book' is then really only a collector's album. An album with accompanying text by Ernst Lewalter in which each owner had to place the pictures by themselves. It stands to reason that Bytwerk was unlucky enough to work with a single copy in which the original owner misplaced the two images.

Timeline of Events

• 1857 — Sepoy Rebellion in India, British victory, and 'methods of suppression'. — No photo, not even a picture close to what's in question exists for the 'blowing of guns', but one photo of two rebells hanging from the gallows is shot by Beato.

• 1882 — Vereshchagin decides to go to India

  [He] did not stay true to the word not to write any more war stories. This time he turned to the subject of the suppression of the British revolt of the sepoys in 1857. The Indian uprising of mercenaries against the colonists employers last but not least been successfully suppressed due to the public executions that received a poetic name "the Devil's wind." In 1857, the newspaper the People's Rares these highly effective interventions are described as follows:

  "The British in India invented a method of execution so horrific that all of humanity is shocked. Are these merciful Christians invented a subtle way – tied living people to the muzzles of the guns, and then fired, ripping people to pieces, spraying blood rain of the pieces of the human body and guts at the audience".

• 1884 — The painter comments on the picture:

  When (two years after the return of the artist from India) painting "The suppression of the Indian uprising by the British" was written, Vereshchagin commented:

  "Modern civilization has scandalicious mainly to the fact that the Turkish massacre was carried out close in Europe, and then the means of committing atrocities too reminded tamerlanovsky times: cut, slit her throat, just sheep. Another thing from the British: first, they were doing the work of justice, the matter of retribution for the violated rights of winners, far away, in India; secondly, did the thing Grand: hundreds tied rebelled against their rule and not of sepoys of the sepoys to the muzzles of guns and no projectile, one gunpowder, shot them is already a great success against pererezali throat or ripping the belly".

  — Andrew Zimoglyadov: "The suppression of the Indian uprising by the British"

• 1939 — The 1857 photo by Beato is printed in the English magazine Picture Post
  _{(June 3, 1939, or not, verification pending!)}

• 1939 — In Munich, Nazis organise an exhibition called "Robber State England", showing the painting by Vereshchagin: , and the photo by Beato

• 1941 — The pictures from the exhibition are assembled for inclusion into a book of the same name with accompanying text from Lewalter. The book is made in two forms: one is a collector's album for cigarette cards as this was the way the pictures were initially distributed, another version has just the finished book with pictures already included. In this book the picture is accurately sourced as being a reproduction of the painting, not a photo.

• post 1945 — Contents of the book end up as translated into English snippets, intended for showcasing German propaganda methods. Sometime at this stage pictures and captions from pages 76 and 77 of the book get confused and that error reproduced. At the latest. Perhaps one collector glued in the picture at the wrong place in the cigarette card collector's book version of the book, perhaps there were misprints.

• ~2008 — Randall Bytwerk uploads the wrong combination of Picture and caption to his page German Propaganda Archive

• Sep 16 '17 at 19:35 — Prompted by the wrong caption (which is also in some archives with wrong meta-data) that now claims the Vereshchagin painting to be 'a photograph copied from a 1939 English magazine', this thread starts.

• June 24, 2020 — The German Propaganda Archive is updated (old version for comparison) to correct the wrong caption, now showing that the picture in question is indeed just a painting by Vereshchagin, (that is: not a photo from a 1939 magazine)