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If you have kitchen units you could put up a chipboard wall (or even MDF) and instead of having battens (not sure if right word?) use the cabinets. Then a small gap and something like dynamat on the existing wall. Like that there would be no bridging.
Decoupling and two heavy materials should do quite some work, but sound from stairs of course also travels through ceilings and floors.
Thank both. Flat is currently without doors, furniture etc, so think it'll be better once it's fitted out but will look to guerilla line the stairs!