Preface:
I looked first for the term "édicule" and verified it function, shape and size. Then I looked for the appropriate translation and found the terms "kiosk", esp. the term "canopy" without referring to any other proposal prior to open the question and remarked that similar terms had already been disagreed being to big. In fact the term "canopy" does not determine the size. In order to prevent an additional disagree I consider it being useful to add this preface, prior to post man proposal.
See below:

1.1 Architecture A projection or shelter that resembles a roof.
‘they mounted the station steps under the concrete canopy’
1.2 The transparent plastic or glass cover of an aircraft's cockpit.
https://www.lexico.com/definition/canopy

2 : an ornamental rooflike structure
3a : the transparent enclosure over an airplane cockpit
b : the fabric part of a parachute that catches the air
https://www.merriam-webster.com/dictionary/canopy

Explanation:
Un édicule est un terme d'architecture qui recouvre plusieurs petites constructions isolées dans l'espace public ou des espaces ouverts, d'emploi et de statut variés.
:
Époque moderne
• Au XVIIIe siècle, en urbanisme et art des jardins, édicule désigne des constructions d'agrément dans les parcs et squares, sans doute plus fermées que les pavillons et kiosques et plutôt d'inspiration antique.
:
• Depuis le XIXe siècle, en urbanisme, édicule désigne de petites constructions dans l'espace public urbain (comme les vespasiennes, des colonnes Morris, des kiosques à journaux ou kiosques à musique, des bouches de métro couvertes, des abris, des volumes techniques). On retrouve l'association aux statues dans certaines, comme les fontaines Wallace.
• Dans les grandes villes, les édicules protègent des intempéries les escaliers comme ceux des bouches du métro.
Édicule moderne de la station de métro De La Savane, à Montréal.
L'un des fameux fosteritos, édicules du métro de Bilbao créés par Sir Norman Foster.
https://fr.wikipedia.org/wiki/Édicule

Édicule d'Hector Guimard, dit La libellule.
• l'accès 3 « Boulevard de l'Amiral-Bruix », constitué d'un escalier fixe coiffé du seul édicule Guimard originel avec marquise de style Art nouveau, se situant face à la place des Généraux-de-Trentinian du côté pair de l'avenue Foch ;
https://fr.wikipedia.org/wiki/Porte_Dauphine_(métro_de_Paris...

• entrance 3 - Boulevard de l'Amiral-Bruix, consisting of a fixed staircase topped with the only original Guimard kiosk with a canopy in the Art Nouveau style, located opposite the Place des Généraux-de-Trentinian on the even side of the Avenue Foch;
https://en.wikipedia.org/wiki/Porte_Dauphine_(Paris_Métro)

marquise f de gare / station canopy o. awning | ~ de trottoir (ch.d.fer) / platform shelter, platform roofing
(Dictionnaire Général de la Technique Industrielle)

TDI : Habillage de cabine ascenseur, édicule ascenseur, tôlerie et charpente métallique
Elle propose notamment l'habillage de cabine ascenseur, l'installation d'édicule ascenseur en inox ou acier, de pylône ascenseur, ou ébrasement de porte
http://www.tdi-tolerie.com/

Those transferring from buses will not be using this new entrance. In fact, this new édicule will essentially be used by residents East of the station and people with mobility issues who need the elevators.
:
Current plan vs current plan removing all the walls inside the proposed edicule.
Imagine all the space and light opened up by using a shared concourse! (1m grid)
http://www.cat-bus.com/2017/03/the-new-vendome-station-long-...

It's a project that Metro announced in April of last year. The plan was to install a vaulted glass and steel canopy over the Dupont Circle station's north entrance, to protect both customers and the escalators from negative impacts of rain, ice, and snow.
https://wjla.com/news/local/metro-bankrupt-contractor-escala...

BART designing prototype escalator canopy to protect customers, employees and machinery
:
BART is in the design phase of a project to install a canopy over an entrance to 19th Street Station in Oakland in an effort to protect customers, employees and the escalators themselves.
https://www.bart.gov/news/articles/2013/news20130627

BART will also install 22 canopies at its downtown San Francisco stations from Embarcadero through Civic Center.
https://www.bart.gov/news/articles/2017/news20171017

The canopy of the lift can be used as an additional parking space or can be designed as lawn. Otherwise one can use paving, tiles, asphalt, concrete or simply metal as floor covering.
https://www.nani.de/products/lifting-equipment/car-lifts-and...

Canopy: The upper platform assembly that travels above the uppermost landing when the lift is raised, and lowers into a recess/opening in the upper landing floor when lowered.
:
6.1.1 Hydraulic Design Carriage and canopy platforms are actuated by four vertical hydraulic ram cylinders which raise and lower the lift platforms within four vertical guide angles attached to the building structure.
https://www.mckinleyelevator.com/app/uploads/2015/10/Vasari-...

The fire progagation was assisted by the 225 square foot ventilation shaft created when the rooftop elevator canopy was cleaned out during standard firefighting operations.
https://books.google.de/books?id=_AZxc6FS8I0C&pg=PA8&lpg=PA8...

Abstract:
The present invention provides improved elevator door hatches. … The door hatch also allows for the drop ceiling to be mounted closer to the canopy, which increases the usable height of the elevator cab.
http://www.freepatentsonline.com/y2018/0105394.html

L'aspect des ascenseurs sans édicule est assez surprenant :
https://www.emrascenseurs.com/details-creation d un ascenseu...

Apparemment, l'édicule est la construction hors œuvre qui émerge du toit et qui sert à loger la partie d'entraînement :
"L’ascenseur / Comment ça marche / Les nouvelles technologies
Contenu

1. La connaissance de l’ascenseur (technologie) :
• Les ascenseurs électriques :
 avec local machinerie (en édicule)
 sans local machinerie (en gaine).
• Les ascenseurs hydrauliques.
• Les nouvelles technologies permettant à l’ascenseur de s’inscrire dans la démarche du développement durable (variation de fréquence avec réjection de courant)...."
https://www.nouvelleviepro.fr/formation/fiche/lascenseur-com...

On trouve une VDI allemande, titre en français :
"VDI 2566-2:2004 Conception acoustique pour ascenseurs sans local de machines" ou, en anglais,
"VDI 2566-2 : 2004 ACOUSTICAL DESIGN FOR LIFT SYSTEMS WITHOUT MACHINE ROOM"

J'ajoute la référence de cette illustration, où on voit très bien l'aspect "cabane sur le toit" - suivre la flèche "Triebwerksraum":
https://www.ditzingen.de/fileadmin/Dateien/Dateien/Feuerwehr...

French term (edited): ascenseur sans édicule

I'm no expert but this seems to be the most widely used term

In USA it would be kiosk, same as at the metro (USA subway). I think this is it (elevator, small villa)

https://www.alibaba.com/product-detail/Glass-Elevator-Small-...

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Note added at 2 hrs (2020-04-27 12:03:48 GMT)
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Kiosk also is a free-standing enclosure for signage, e.g., at an elevator: https://touchsource.com/products/

Same word as metro kiosk: https://gvshp.org/blog/2012/10/04/historic-station-sequel/

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Note added at 23 hrs (2020-04-28 08:31:22 GMT)
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Tony M, I should have been clearer. I was suggesting that if kiosk is a synonym for edicule as a hut (per Asker's original suggestion), that's a possible source of confusion because a kiosk in EN may be two entirely different things which both coincidentally are found around elevators: the hut as on the Budapest/NYC metro, or free-standing encased signage which can be an elevator signaling panel separate from the wall but has nothing to do with huts. In fact I'm not even sure which of the two things the ST is referring to. Many FR-captioned pictures of elevator edicules seem to involve both kiosk huts and signage kiosks.

Is what the firemen seem to call it...

https://books.google.co.uk/books?id=i4bmE2TqY6sC&pg=PA338&lp...

https://books.google.de/books?id=6lgsMTl3IfQC&pg=PA52&lpg=PA...
....the little 'hut' that sticks up....

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Note added at 1 day 1 hr (2020-04-28 11:10:53 GMT)
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Something that looks similar(ish) to this: