 Solution

There is a conflict in the printed question: with dynamical $X$ and the displayed superpotential, $F_X=\tilde{\varphi }\varphi =M$ forces $M=0$, so the $N=1$ theory has no vacuum branch parametrized by nonzero $M$. The natural intended calculation is the Kähler quotient of the $N=1$ D-flat SQED matter fields before imposing that extra F-term.

On this quotient, $|\varphi |=|\tilde{\varphi }|$ and the gauge-invariant coordinate is $M=\tilde{\varphi }\varphi$. Hence $|\varphi {|}^2=|\tilde{\varphi }{|}^2=|M|$, and the canonical Kähler potential descends toThe associated Kähler metric isIt is smooth for $M\ne 0$ and has a conical singularity at $M=0$, locally $\mathbb{C}/{\mathbb{Z}}_2$. Physically, the charged fields become massless and the U(1) gauge symmetry is restored there, so integrating out the vector multiplet to obtain a sigma-model metric ceases to be valid.