= Solution
In the <sudden approximation>, the nuclear charge changes before the electronic <wavefunction> has time to evolve. Immediately afterward the state is still the original hydrogenic 1s <wavefunction>, now expanded in the eigenstates of the new Coulomb Hamiltonian. The final-ground-state <probability> is the squared overlap, rather than one minus an <energy> change.
For initial and final charges $Z_1,Z_2$, the normalized 1s overlap is
$$
\langle1s,Z_2|1s,Z_1\rangle
=\frac{4(Z_1Z_2)^{3/2}}{a_0^3}\int_0^\infty r^2e^{-(Z_1+Z_2)r/a_0}\,dr
=\frac{8(Z_1Z_2)^{3/2}}{(Z_1+Z_2)^3}.
$$
Using $Z_1=1,Z_2=2$ gives
$$
\boxed{P_{1s}=\left(\frac{16\sqrt2}{27}\right)^2=\frac{512}{729}\simeq0.7023.}
$$
The remaining <probability> includes higher bound $ns$ states and the $s$-wave ionization continuum. In this central, instantaneous model the initial orbital has $l=0,m=0$, and angular orthogonality excludes states with $l\ne0$. The electronic spin is unchanged. Nuclear recoil and interactions with the emitted decay particles are outside this leading sudden-charge-change approximation.
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