The initial point of a vector is the point that the vector starts from or originates from. To find the initial point of a vector, we need to subtract the vector from the terminal point of the vector.

In this case, the terminal point of the vector is given as $B(2,0)$, and the vector is given as $\mathbf{u} = (1,2)$. To find the initial point, we simply subtract the coordinates of the vector from the coordinates of the terminal point, like this:

$$\begin{aligned} B - \mathbf{u} &= (2,0) - (1,2) \ &= (2 - 1,0 - 2) \ &= (1,-2) \end{aligned}$$

Therefore, the initial point of the vector is $\boxed{(1,-2)}$.