Imagine an ellipse $$1 = \frac{1}{4}x^2 + y^2 $$
With focus one at $$F_1 = (-1,0)$$
Consider the following coordinate transformation:
$$x = 2u$$
Substituting gives
$$1 = \frac{1}{4}(2u)^2 + y^2 = u^2 + y^2 $$
The focus is then at in this new coordinate system $$F_1 = (-1/2, 0)$$
Is the above reasoning right?
Is there also a way to transform one of the foci to the centre of the circle?
In general, transformations of this sort do not preserve foci.
Starting with your ellipse $$ 1=\frac{1}{4}x^2+y^2=\left(\frac{1}{2}x\right)^2+y^2, $$ you applied the transformation $u=\frac{1}{2}x$ and $v=y$. This type of transformation is called a linear transformation as it can be represented by a matrix product: $$ \begin{bmatrix}u\\v\end{bmatrix}=\begin{bmatrix}\frac{1}{2}&0\\0&1\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}. $$ Your transformed ellipse is the circle $1=u^2+v^2$, a circle centered at the origin of radius $1$. This circle has both foci at the origin. The foci of the original ellipse are $(\pm \sqrt{3},0)$, which map to $\left(\pm\frac{\sqrt{3}}{2},0\right)$ under this transformation, neither of which is the focus of the new circle.
In general, applying a linear transformation to an ellipse will result in a new ellipse. However, the locations of the original foci do not provide much information about the locations of the new foci for the new ellipse. The center of the original ellipse, however, maps to the center of the new ellipse.
Using an invertible linear transformation, there is no way to transform the foci of a non-circular ellipse to the center of an ellipse or circle.