This is such an i nteresting question. You can be L- or R- handed at many different tasks. You would not be left handed at everything. Our brains develop over time, and we master different tasks at different times.
So you may be left-handed at certain tasks by a certain age, and right-handed at other tasks - the handedness emerging as and when each task is mastered.
If you think of drawing with the left hand as an indicator of handedness, then you could say this is settled at the time drawing is mastered. If you think of writing with the left hand as an indicator of handedness, then you could say this is settled at the time writing is mastered. It all depends on how you define handedness. Because it is not black and white (different hands may be preferred for different tasks), it is hard to say when it is actually settled.
Also, our brains are highly adaptable and handedness can change over time with practice. And of course, some people can do some things equally as well with either hand (like lmsunshine's son)
I think it is an unanswerable question unless we are specific about which activities we mean.
I think it is a fascinating area because it also tied up with superstitions. Even the word for 'left' has negative associations (eg it is sinistro in Latin, something like gauche in French which means clumsy.awkward). Over time, people have been discouraged from using their left hands so these people have had to adapt.