Not a local, but do work for a design studio with a London-based HQ. They’re based in Shoreditch, which is quite a nice center of art and design (and pubs). Have to be at least two-dozen other furniture/art/design studios in the area, as well as a plethora of street art and local shops.
Hi Joe as an ex local for design you should definitely head to East London for the Art and Design scene. Shoreditch is a fun and trendy area which is also close to Clerkenwell Road which houses a lot of the London arm of most design firms. The Vitra store is on Clerkenwell and also the IDEO London studio.
You should definitely check out the Design Museum - they’ll be the closet to having a collection focussed on ID. The Geffrye museum also documents the home and lifestyle changes of the last 400 years. Definitely head to the Conran shop like Michael/Yo said.
Although a little pricey Sketch has interesting decor if you want to head there for drinks. The bathroom is the most interesting with the cubicles being egg shaped pods placed around the room