Which computer controls auto thrust on an Airbus A320?
More specifically on your question:

See diagram below:



In the autothrust mode (A/THR function active), the FMGC computes the thrust, which is limited to the value corresponding to the thrust lever position (unless the alpha-floor mode is activated).




  So as you can see computers/controllers that control the A/THR are:
  
  
  FCU
  FMGC/S
  EIU or EEC
  FADEC
  ADR
  


The **speed or Mach target may be:


  ‐ Selected on the FCU by the flight crew
  
  ‐ Managed by the FMGC.


EPR mode is the normal mode to control the thrust. The required EPR is set by controlling the fuel flow.

DSC-70-35-20 P 1/2

The FADEC has two mode of power setting, EPR and N1 modes.
The FADEC computes the command EPR as a function of : 

‐ Thrust Lever Angle (TLA)

‐ Altitude

‐ Mach number

‐ Ambient temperature

‐ Air inlet total temperature (T2) ‐ Service bleed.

Arming conditions of the A/THR are numerous. The following is a list of the most important ones:

‐ One FMGC operative

‐ One FAC operative

‐ Two ADIRS operative

‐ Two FADECs operative

‐ One channel of the FCU operative

‐ One LGCIU operative

‐ A/THR is not manually disabled (instinctive disconnect pb has not been pressed for more than 15 s).


  So to summarise:
  FMGS computes the THR and the FADEC computes EPR and through EIU/EEC and ECU commands the fuel flow to the engines in order
  to give the desired EPR.