= Solution
The <Abelian Higgs model> has local $G=U(1)$, and a nonzero minimally charged scalar expectation value has trivial <stabilizer> $H=\{1\}$. Thus $M\simeq U(1)\simeq S^1$ and $\pi_1(M)=\mathbb Z$, while $\pi_0$ and $\pi_2$ are trivial. The defects are local strings, or <Nielsen-Olesen vortices>, whose winding fixes flux $2\pi n/e$.
Put $R=\eta+\phi/\sqrt2$ and $\vartheta=\psi/(\eta\sqrt2)$. The specified gauge-field shift is $A=A'+\partial\vartheta/e$, so
$$
D_\mu\Phi=e^{i\vartheta}(\partial_\mu R-ieA'_\mu R),\qquad
|D\Phi|^2=\frac12(\partial\phi)^2+e^2R^2A'_\mu A'^{\mu},\qquad F(A)=F(A').
$$
Expanding the potential gives $V=\lambda\eta^2\phi^2/2+O(\phi^3)$. Hence the <broken-phase spectrum of the Abelian Higgs model> follows from
$$
\mathcal L_2=\frac12(\partial\phi)^2-\frac12\lambda\eta^2\phi^2
-\frac14F'_{\mu\nu}F'^{\mu\nu}+\frac12(2e^2\eta^2)A'_\mu A'^{\mu}:
$$
$$
\boxed{m_\phi=\sqrt\lambda\eta\quad\text{one real scalar},\qquad
m_A=\sqrt2e\eta\quad\text{one massive vector}.}
$$
The phase is not an additional physical massless scalar: the <Higgs mechanism> supplies the vector's longitudinal polarization. In the symmetric phase the <complex scalar> has two real <degrees of freedom> and the massless vector has two polarizations, a total of four. In the broken phase there are one scalar and three massive-vector polarizations, again four. The symmetric phase is the thermally restored phase; the origin of the given zero-temperature potential is unstable, not a separate stable particle vacuum. The local polar change of variables cannot remove global vortex winding everywhere, so the massive spectrum is consistent with string defects.
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