What is the vertical speed limit when altitude mode is engaged?
What you are describing is the managed climb mode of the Airbus, where the Flight Management Computer (FMC) determines the optimum climb speed and the autoflight system will then pitch to maintain this speed.

From the A320 FCOM (Autoflight - AP/FD Vertical Modes):


  Climb Mode (CLB)
  
  CLB mode guides the aircraft in a managed climb, at either a managed or a selected target speed, to an FCU selected altitude, taking into account altitude constraints at waypoints. The system also considers speed constraints if the target speed is managed. [...]
  
  The pilot can arm the CLB mode during the takeoff, go around, climb and cruise phases and engage it during the climb and cruise phases.


The vertical speed is therefore determined by the difference of the current and the target airspeed. All protections like stall protection based on angle of attack still apply, which may limit the actual climb rate.

The same logic also applies when descending instead of climbing, where the aircraft will maintain a given descent speed by pitching at idle thrust instead of climb thrust.

Once you get close to the selected altitude, the ALT* mode will take over, which follows a vertical speed instead of pitching for speed:


  Altitude Acquire Mode (ALT*)
  
  ALT* (green) mode guides the aircraft to acquire the FCU selected altitude.
  
  ALT* (magenta) guides the aircraft to acquire an altitude constraint provided by the Flight Management.
  Once the aircraft has reached the altitude, the altitude mode (ALT green or magenta) engages.
  
  Engagement Conditions
  
  The mode engages when the aircraft reaches the altitude capture zone defined by the aircraft vertical speed
  (among other parameters).
  
  [...]
  
  Guidance
  
  The ALT* mode has internal V/S guidance that is a direct function of the difference between present altitude
  and altitude target. This V/S internal guidance has priority over the target speed. Therefore if a thrust reduction
  occurs in ALT* (Engine out), the vertical speed order will not be modified and a speed variation will occur. The ALT*
  mode does not have a mode reversion that results from speed variation.
  
  [...]
  
  ALT* and ALT CSTR* modes have internal protections that decreases the vertical speed when VLS or VMAX is reached (VLS or VMAX becomes the priority target).
  
  [...]
  
  The system switches automatically to ALT (altitude hold) when the altitude deviation becomes less than 20 feet.


The vertical speed is therefore only limited by the internal guidance logic, but will be overwritten when VLS or VMAX are reached providing the usual flight envelope protections.