How soon can I start my turn after reaching the missed approach point, when missed approach procedure calls for a climbing turn?
Based on your question and the circumstances you indicate:

On an approach that takes you down to 200' HAT (height above touchdown) and specifies a climbing turn without a "climb straight ahead to [some altitude] then climbing right turn," you should climb to 400 feet AGL and then start your right climbing turn along the missed approach course track.

From FAA Order 8260.3C, Section 2-8 Missed Approach, paragraph 2-8-6 [pertinent portion]:


  "2-8-6. Turning Missed Approach Area. If a turn of more than 15 degrees from the FAC is required, a turning or combination straight
  and turning missed approach area must be constructed.
  
  Note: If the HAT or HAA value associated with the DA/MDA is less than 400 feet, construct a combination straight and turning missed
  approach (see paragraph 2-8-8) to accommodate climb to 400 feet above
  TDZE/airport elevation prior to turn."


========================================================================