Can anyone confirm the identity of these WWI uniforms?

For ease of communication, the people will be referred to in the following pattern:

     A   B
   C   D   E

Final Conclusion

1. A-D are Officers of the Machine gun Corps of the British Expeditionary Force.
2. E is a Portugese officer

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The Chase

That photo and the information available surrounding it makes a tough nut, the following is therefor only a preliminary partial answer.

What we can gather from this photo:

1. These are officers posing for a photo on the western front of the First World War. It looks like the season is either early spring or late autumn (the gloves on D and the lack of heavier overcoats, plant surroundings and lighting). They all wear a British Pattern Uniform. A Sam-Browne-belt that is only slightly different on E (the others are fastened with a knob whereas E has a loop). All wear puttees around their boots. But giant feet B wears ankle boots whereas C and D have different foot wear, C possibly riding boots, matching his riding pantaloons ("riding breeches" with a patch on the inside seam, resembling a stripe). Riding patches on pantaloons is no proof for cavalry membership or some other necessity, other than style or possibly comfort:

   Haig with army commanders 1918 IWM Q 9689.jpg (Front row donning cavalry patches. Note the variety of shirt colours and even differences in necktie style.)

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2. B, C and D have insignia on their peaked visor service dress hats. This insignia seems to belong indeed to the machine gun corps and is apparently missing on A's hat, and whether it is just the lighting or different material, this badge gets somehow darker from D to C to B. Note that the hats for a A, B and D seem to be the service dress hat, being quite stiff and still well shaped. The hat on B seems to be either quite worn or of the trench hat variety that is a bit more flexible. Difference between service hat and trench cap, typical British officer jacket:

   
   From: Jane Tynan: "British Army Uniform and the First World War: Men in Khaki", Palgrave Macmillan: London, New York, 2013, p78.

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3. The badge:

   

   150. The Machine Gun Corps. Cap badge, 1915–22. –– 154. The Machine Gun Corps. Officers' collar dog. Bronze.
        From: F. Wilkinson: "Badges of the British Army 1820 to the Present. An lllustrated Reference Guide for Collectors", Arms and Armour: London, 1997, p 62.

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4. The hat on E is very unlike any hat for armies of the Eastern front Serbian hats, like most hats of "Slavic" armies were much "pointier" in several ways. The hat on E most likely resembles more the British forage cap, being quite flat, although the colour seems to be quite off on this one. That may be due to a different colour cover. The visor and an optional pummel on top seem to be less visible here just because of the angle. This of course only if it is indeed a type of forage hat. Another possibility is that especially in colder weather the head gear got even more flexible to choose from:

From: Peter Hart: "Fire and Movement. The British Expeditionary Force and the Campaign of 1914", Oxford University Press: Oxford, New York, 2015.

If it is a type of forage hat, the versions appearing after 1900 are likely candidates:

(British forage cap, turn of the century, styles changed quickly and were never universal across all units, so forgive the fancy pattern here.)

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5. The most intriguing facial beard styles on display here are the clean shaven faces of B and C, the little moustaches on A and D and the "old" style wide moustache on E. All except E might point to a trend that of facial hair style that emerged after tight fitting gas masks were referred over stylish beards. This makes a date of "after 1915" more likely.

In 1916, the Military Mail announced that the traditional army custom of wearing a moustache would be brought to an end: ‘That the moustache should now no longer be compulsory in the Army may come as a surprise to the older, and as a relief to some of the younger, members of the Service’; the article also lamented that it had been ‘the mark of the British soldier.’ (Khaki, p35.)

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6. The shoulder insignia on A, B, C, and D seem all to differ, indicating quite different units the soldiers belong to. E Has further no corps insignia on his lapels but a triangular badge that is not for me to make out in this resolution.

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7. The shirts seem to be of quite a lighter colour on C over all and just the collar on E. But the universal use of collar pins (or collar bars, collar clips) under their neckties would place all of these officers firmly in the British or Empire forces. No other army of that time seemed to have this preference for an arching tie sticking out of the uniform.

But again, shirt colour alone is not as significant as one might think:

From: William Langford: "Images of War. Great Push. Battle of the Somme 1916", Pen & Sword: Barnsley, 2012.

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8. The animals on display, two cats on the shoulders of A and B and a dog on the lap of D seem of as little consequence as the dark pipe in D's mouth or the light pipe in E's hand. (See "update 2" for the likely identity of this accessory.)

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9. One effect visible seems to be either the front line degradation of uniform dress regulation or the rag-tag reality of difficulties in supply:

photo from 'Punch' periodical of 1914. via: Remembering the fallen: Officers

This is the main cause for most of the non-uniformity issues on display here: officers were in part expected to purchase their gear from their own money but were also given even more leeway in style compared to frontline duty enlisted man. Some basic regulations had to be observed, but the exact cuts, materials and sometimes even styles were widely mixed to an extent. This is especially true for either frontline conditions or off-duty times beyond the trenches.

Given that wartime visual culture sought to recruit men through the seductive image of a smart military uniform, the failure to supply enough uniforms was a delicate issue. The War Office did not obtain enough khaki uniforms in the opening weeks of the war, and many early recruits were forced to wear replacement uniforms, which became known as ‘Kitchener blue’. The uniforms were obtained from a variety of non-military sources: 500,000 suits of blue serge uniforms from post office stocks and approximately 500,000 greatcoats purchased from the clothing trade. The War Office ordered a further 1,300,000 jackets and pairs of trousers as well as 900,000 greatcoats from Canada and the United States. (Khaki, p47.)
Traditionally, the typical army recruit was issued with a regulation uniform of average quality and cut, while the officer was expected to buy a made-to-measure suit. Officers had more choice about where they could procure their uniforms. Dress regulations for officers included illustrations of actual garments, a practice well established by the First World War and reflected in the design of the 1904 and 1911 publications. As argued elsewhere, images of discipline were key to the ideological construction of soldiers. (Khaki, p66.)
For British army officers, getting fitted out with a uniform usually involved a visit to their tailor, but when this established practice was broken, many men purchased items from clothiers. In turn, many clothiers and outfitters wanted to benefit from the modern image of khaki and so sought to capitalize on the traditional link between officering and tailoring. (Khaki, p124.)
Initially the army was reluctant to accept anything departing too visibly from regulation, but according to minutes from the directors’ meeting of the Quartermaster-General Department of 29 August 1914, Kitchener was not too concerned, taking the view that as long as men in individual units dressed alike, the improvised outfits would be adequate in the short term. (Khaki, p47.)

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Conclusion All this taken together might indicate that this picture was taken more in the later part of the war. It displays at least four officers from the British Expeditionary Force, from different units within the machine gun corps and one that might be from other Imperial forces, if he is not also in the Royal Army. Although this represents a lot of educated guess work and indirect evidence, for A–D the inferences presented so far seem adequate. Soldier E remains a bit more elusive.

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Update on Mr E:
He might indeed be French after all. It is not only the more impressive moustache that would indicate his membership. While virtually all French uniforms of that era are depicted either with a standup collar, sometimes like a mandarin collar, or a very wide falling collar, there were apparently some instances of imitating the British.

(From: Ian Sumner & Gerry Embleton: "The French Army 1914–1918", Men-at-Arms Vol 286, Osprey Military, 1995, p6.)

The undecorated and very flat cap might then be a kepi and the slightly longer jacket would also be a slightly better match. The lapels however are even wider on E than in this picture. However, the cut of the uniform still points more likely towards British, New Zealand, Australian, South African or Portuguese armies than French.

But all these possibilities are still more likely than a Uniform of Russian, German, Austrian, Italian, Romanian, Polish, Bulgarian, Serb, Belgian, Greek, Japanese origin. Unfortunately uniforms in actual use seemed to be less uniform than often expected.

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Update 2:

One source of this picture in a printed book seems to be:

Found by Lars Bosteen in: Neil R. Storey: "Animals in the First World War ", Bloomsbury: London, Oxford, 2014, unpaginated/ebookpage69.

Solving the national identity of Mr E. In this versiopn of the picture the Portuguese officer seems to carry a quite large swagger stick. So large that in fact it looks more like a vine staff or pace stick.