Additional pics are needed. Maybe take some from the bottom and top of the stairs.

How tall are you? 5'2" me has a different idea about navigating storage above stairs than my 5'10" son in law.

What is it you plan to store in the "wasted" space? What is if F.O.R ..........before you feel compelled to use the air?

Apologies here are some better images. does anyone have ideas of what could be possible?

"Possible"

is the determined result of a need, a purpose.

You aren't even answering the question, you are just determined to use something...for WHAT?

Get a contractor in your home, AFTER you figure out what it is you want to store. He won't be able to read your mind, either.

Jan’s image above seems like an option - although if it were me, I’d wait and live in house before a project like this - it’s a narrow staircase so hard to imagine wanting to reduce any perceived volume

(totally a me thing but if storage was needed in the home, I would add it a lot of other places before considering something over the stairs)

Steady on jan, looking for some ideas and inspiration not an argument. understand the opinion but it’s horrible looking imo. Feel it’s best to do the work now while the whole house is being rennovated.

we don’t have much storage so we could see we need that?

Can you post a photo from the middle of the stair case looking up so we can see the landing area? Where is your attic entrance?

You mean the attic space ???? The rest of the space above the staircase is for head room. Attic spave usually great for storing stuff you rarely use like luggage, Christmas or other seasonal decor and sometimes even sports eqipment not used in ceratin seasons . Anything is possible so see theguy with the harness working in a space like that how good are you at wearing that to access stuff ? Use the attic put a fab light fixture in that space and be done

I'd think about how you'd design a staircase going up to an imaginary 3rd floor, then use that space for storage. It will look the most normal to build the storage at a slope parallel to the current staircase. Nobody will hit their heads. The last pic that Jan posted seems the best to capture this idea. It truly looks like an enclosed staircase to a 3rd floor.

Are you in the UK? Building codes differ by country.

If it were me, I’d have one or two pro’s come in to suggest/quote, as there are a lot of factors to consider structurally and of course cost wise. Also consider: the effect visually as it could feel a bit oppressive (from both floors), and the actual functionality/ease of using.

If storage is the driver vs closing up the expanse, consider using the attic if feasible. Then add ceiling lighting and large art (or wallpaper) on the staircase to add interest.

What about making the landing wider at the highest ceiling point and having a wardrobe there?

For the question of whether you should add storage over the flight of stairs, a single photo is all you need:

See the existing angle over the last 2 stair treads? That is painstakingly framed to provide a clearance of 6'-8" (+ or - if you are not in the U.S.) clear over the stair tread directly below it, and that must remain all the way up the flight.

The "air" over the stair flight actually helps to open up the second floor visually and provide a feeling that you are not in a tightly controlled maze. The hallway is a collection and congregation point of circulation from the rooms which benefits from feeling more open.

Surely better located/accessed, more functional, and more matching-to-current-level-of-aesthetics storage can be found somewhere in the house.

^^^

Especially that

What landing, where for a wardrobe? You have no landing that I see. Just hallway and steps.

I wonder if you are maximizing layout in all of your rooms and getting the most storage in that that you can get.