There are two large coffee shops across from John Lewis, one on top of the other (ie same location on different floors, you can see them both from the opposite escalator balcony) with plenty of seating.
A lot of these places are noisy; the centre is noisy. The back of Starbucks (not the one in the main atrium, the one in a unit), quite near to these two, is quieter, but it's not very big so you might get moved on unless you buy plenty of drinks. No idea on sockets as I don't work from there, just visit sometimes.
There are also restaurants that if not busy might let you buy some lunch or a snack, coffees etc., and work from there, worth having a look. Not the ones inside the centre (noisy) but the ones outside in a row (with a break and steps in the middle) along the walkway from one main entrance to the other.
The location around and in Five Guys is sometimes a quieter spot in the centre than some of the restaurants and cafes are, but as it is school hols and a major hangout for the youth of the area, doubt it, and it's near the cluster of child-related shops too.
There is a Regus office place by Shepherd's Bush tube station, if it transpires you actually aren't allowed to work from a cafe or it is too noisy.
Black Sheep and Costa are not in the Westfield Centre, they are on the walk-up from the underground station on the same side, effectively within the outdoor bus station with the overground station opposite. And in the same location as Regus. (There might be another Costa inside the centre, I can't remember. There might also be some other cafes, ditto.)