For a theistic dualist, the moment when the brain is no longer just a lump of living neurons but a conscious mind is the moment their deity of choice puts a soul in the body.
For an identity physicalist, who believes that the mind and the brain are one and the same, the question isn't problematic either. The mind starts to appear the moment the brain starts to develop.
This leaves us with non-theistic dualists, who believe that the mind and the brain are two different types of thing, but still believe that the mind can be explained by the laws of nature.
It seems to me that any non-theistic dualist solution to the mind-body problem would have to account for how to differentiate between a living but still non-sentient brain (an embryo, a person in a coma, an animal too low in the hierarchy to be considered sentient, etc...) vs a brain that actually supports a mind.
The details are obviously more the province of neuro-science than of philosophy, but I still feel that a non theistic dualist would need to give some general criteria for when brain states lead to mental states and when they remain just brain states.
Has anyone of note addressed this question: From a non-theistic dualist point of view, when are brain (neural) states alive but inanimate and when do they subsume mental states? At what point does the emergence of a mind occur?
[ I was originally going to use the term property dualist, but realized that property dualist and non-theistic dualist might not be the same thing. ]