Ashes to diamonds --- How?
According to US published patent application 20030017932 "Cremeting; filtering carbon particles; sublimation; polishing"


  The preferred process for collection begins with the oven operator positioning the body in the oven so that the head and chest area are not positioned directly underneath the main burner. This can be accomplished by positioning the body to the left or right side of the main burner, or positioning the body so that the legs and feet are underneath the main burner rather than the head and torso. Positioning the body in this manner assures that carbon will remain in the body's head area. The carbon can then be gathered by hand, or by using a metal shovel or scoop, or the like.


So, in this technique (or proposed technique) the idea is to not let the normal cremation process occur, which would change much of the carbon to CO2.  It seems more like pyrolysis to charcoal.  

They go on to say:


  Another alternative embodiment for carbon collection involves collecting carbon from pulverized cremated remains. These remains consist mostly of ash, but depending on how the cremation was performed, there may be traces of carbon particles mixed in with the ashes. The remains can be placed in a vacuum induction furnace. The furnace is heated to 2000 degrees centigrade in a vacuum ranging from 30 toir to 500 torr. Chlorine gas is injected into the furnace, and reacts with the impurities to form chlorides. The impurities leave the carbon in the form of chloride gases, and are filtered as they exit the furnace.


For more possible methods read this and other patent applications such as US20040031434.


  In another alternative embodiment of the invention, the remains can be cremated conventionally, mixed with additional carbon from another source, and purified as described above. It is contemplated that, using this technique, a gem containing at least some of the original carbon from the cremated remains can be prepared, even if the amount of carbon present in the remains alone is insufficient to make a gemstone of desired size or type.


There is no way to say what actually occurs at a particular company, unless someone has first hand knowledge, which I don't.