Shorts Rochester Airport Works was taken over by Elliott Automation at some point after Shorts vacated it (not sure when, sometime in the 1950s I think), Elliott Automation then became part of Marconi, which became part of GEC and then went to BAE Systems on its creation. The original Shorts hangar is still at the core of the production facilities, but much of the engineering development work is in GEC-era office blocks built around it.
The Shorts Flying School site at the other end of the airfield was at one time the Pobjoy works IIRC, it was a training facility for customers for a long time, but lost that role in the '90s and was demolished about 10 years ago.
The associated research centre in the middle of Chatham (whose proper name I forget - architecturally notable for having a board room built as a sort of flying bridge/penthouse) was sold off and turned into an apartment block in the 90s.
I'm not sure that anyone ever did anything with the Seaplane Works on the Esplanade after Shorts upped sticks. ISTR parts of the site were still extant when I moved here in the mid-80s, but that area is all modern housing developments now.