I got my current chair nearby, possibly where OP is talking about. I wanted to be able to sit and adjust chairs and needed one that would tilt forward. Their selection is a bit chaotic and didn't have the chair I really wanted, but I liked being able to try out a few.

I think you'll have similar success watching Gumtree, craigslist, etc.

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Andrews Office Furniture - Old Street & The City 250 - 254 Old St, London EC1V 9DD

Yo that's the one!

Haha small world! I went to the kebab shop across the street on a regular basis when I lived nearby for Techstars in 2015

Nearly every city has one. Every office needs some form of desk and almost always a chair. These are also worth repairing in enough cases that it is worthwhile for some small local company to deal with that, but they are not worth repairing often enough that any company would keep that repair person on staff (most chairs are scrap when they break). So nearly everyone contracts with a local company to buy office furniture and keep it repaired. (where I work there are 10 extra chairs in a back room, when your chair breaks you trade it for one in that room, if you hire someone you grab a chair. Then they come once a week and count chairs, replacing any broken ones, and adjusting our chair lease if there are more/less than 10 chairs.)

Finding this company shouldn't be too hard, but they don't have a nice retail location since most of the business is they bring chairs to you for an extra fee. However they always have a location and generally they have used chairs for cheap they are happy to sell you.

Andrews Office Furniture, next to inmarsat?