My take - both the controller and the pilot on the radio were being needlessly snippy toward each other.
The low altitude airspace around JFK can be very congested at times (limited options for vectoring) and arrival flows tightly spaced to maintain expected arrival rates. It is a very challenging piece of airspace to work.
The JFK final controller was trying to manage a flow of traffic based on the expectation that everyone could hold 180 KIAS to 5 DME. Maybe not the best plan in the world, but I'm guessing that they use that plan often enough, and crews (mostly) comply, so that was the plan for the moment.
British Air said they were unable to maintain the requested airspeed. Fair enough. At that point, the controller needs to either slow everyone behind BA down, possibly causing a domino effect back to the previous controller, or pull BA out of the sequence and reinsert them with additional spacing. That decision had to be made in fairly short order, however, the radio calls to work this out could have been handled a bit more diplomatically by both sides.
The proper place to have resolved this dilemma was on the previous frequency. The need to maintain a certain speed to a certain DME could have been inserted into the ATIS broadcast or passed on by the first approach controller at which time the proper spacing could have been worked out. By the time it wound up in the final controller's lap, the options to fix it were limited.