Who decided that the Boeing 737 MAX planes that were airborne when the grounding was issued cannot enter and land in EU air space? On March 12th, 2019, EASA grounded the Boeing 737 MAX series by publishing their Emergency Airworthiness Directive 2019-0051-E (AD): 


  Effective Date: 12 March 2019, 19:00 UTC
  
   [...]
  
  Required Action(s) and Compliance Time(s):
  
  From the effective date and time of this AD, do not operate the
  aeroplane, except that a single non-commercial ferry flight (up to
  three flight cycles) may be accomplished to return the aeroplane to a
  location where the expected corrective action(s) can be accomplished.


The PDF document was created at 17:12 UTC and went into effect practically immediately. The AD does not specify any particular instruction for actions concerning flights already airborne. 

Of course there were a number of long-haul flights en-route, that clearly started before the publication, and were scheduled for landing in the EU hours after the effective date/time of the AD. Closely related questions have already been asked on Avitation SE: 


What happens to an airborne plane when its type is grounded by an authority? 
Why would a flight no longer considered airworthy be redirected like this?  (Smart Wings QS 1201 was scheduled for DXB 15:30 UTC - PRG 23:20 UTC)


After March 12th, many people were convinced that EASA has forbidden entry or landing for the 737 MAX in EU air space. I cannot read that from the directive or any other reliable source. 

We all agree that operation of a plane shall not be finished mid-air. The AD could not be followed literally for flights like QS 1201. Unless this case is covered by a different applicable document, EASA failed to give a clear directive.
Emergency ADs shall always serve the purpose of aviation safety. I cannot see how diverting numerous flights on very short notice can be safer than allowing landing at the planned destinations. 

For comparison, this is how the FAA worded their AD on March 13th in this corresponding part:


  This Order is effective immediately. This Order prohibits the
  operation of Boeing 737 MAX airplanes by U.S. certificated operators.
  This Order also prohibits the operation of Boeing 737 MAX series
  airplanes in the territory of the United States. Boeing 737 MAX series
  airplanes covered by this Order, if in flight at the time this Order
  is issued, may proceed to and complete their soonest planned landing,
  but may not again takeoff. 
  
  Special flight permits may be issued [...] including to allow non-passenger carrying flights, as needed, for purposes of flight to a base for storage, production flight testing, repairs, alterations or maintenance.


FAA's AD covers the case of airborne 737 MAX explicitly. On the contrary, the EASA AD requires interpretation at this point, and therefore this is my question:

Who had the right and duty to interpret this AD and to decide what should happen with the B737 MAX in flight? 


Was it ATC in the role of an executive authority? 
Was it the Airline that is risking its certification or wanting to minimize storage fees? 
Was it the PIC who is risking a hefty fine?
Or am I missing something entirely?
The operator's airworthiness department interprets the AD and informs the operations department and senior management.  Ops decide how to deal with airborne aircraft. 

The EASA AD actually had more generous continued flight provisions than the FAA issued AD.  It was in no way intended to prevent aircraft in flight from landing.  Airworthiness would immediately talk to operations.  Operations would decide what to do, though in most cases a few phone calls would be made to senior management.  If the aircraft can't make it back to a 'home' port, that pretty much dictates what is going to happen next (continue to destination).  In the subject case, a number of aircraft were airborne and capable of returning to a home port.  So you need to figure out what to do next - continue flying to your EU destination and get your passengers there with minimum inconvenience (with possible adverse publicity for not discontinuing the flight as soon as possible), or simply get the aircraft back to base so that you don't have it stuck somewhere needing maintenance (and reading between the lines, Ops and Airworthiness would be concerned that the recovery might become more complicated than the 3 flight cycle limit ferry suggests).  So better to get your pax down ASAP, at a port where you can easily carry out whatever rectification is needed, and importantly too, also one where you have more aircraft available to get your passengers to their destination without having to do positioning flights. 

In some cases, such as TUI, they elected to divert the aircraft to the nearest suitable port, regardless of destination or home port convenience.