A similar question could have been asked about electromagnetism before we knew about QED. But it was well understood how electrons, described by the Schrodinger equation, moved in response to an electric field: you insert the classical electromagnetic potential as the potential $V$. Nobody was saying "We can't know how the electron responds to the electric field of the proton... because we don't have a theory of quantum electrodynamics."
We can just use classical Newtonian gravity
In other words, you only need quantum gravity if you're considering a quantum mechanical situation where the effect of gravity is not approximated as a potential which is a function of the electron's position. For example "What is the potential between the proton and the electron when they are within the electron Schwarzschild radius". The specific question you're asking about is okay with that approximation.
For example, we can be sure that the effect of gravity on an electron in a hydrogen atom is given by perturbation theory, with the perturbing Hamiltonian being the gravitational potential between the proton and the electron.
It's experimentally verified even
Consider also the experimental observation of quantum states emerging within a gravitational potential (link to abstract page). Ultracold neutrons bounce against some kind of solid and their distribution in observed heights is consistent with the quantum mechanical calculation, and not the classical one. This proves experimentally that the Schrödinger equation is valid even if the potential in question comes from gravity.
It's just quantum mechanics with a gravitational potential
So to answer your specific question let's just see what the Schrodinger equation plus the postulates of quantum mechanics say. We have a test mass $t$ in a wavepacket a distance $R$ away from a hydrogen atom. It is neutral and only responds to gravity, so it moves in a potential given by
$$
V(x_t,x_e,x_p)=-G\left(\frac{m_em_t}{|x_t-x_e|}+\frac{m_pm_t}{|x_t-x_p|}\right)
$$
The whole system evolves under the Schrodinger equation
$$
i\hbar\frac{d\Psi(x_t,x_e,x_p)}{dt}=(V+K)\Psi
$$
Where $V$ is the potential (including gravity!) and $K$ is the kinetic energy.
And if you do a measurement on $\Psi$, you project onto the subspace of the Hilbert space which is consistent with the result of your observation (please don't debate interpretations of quantum mechanics in this comment section - yes I know in your favorite QM interpretation this event would be phrased differently).
So after all that text, here's your answer
After you measure the electron's position, you project onto the subspace consistent with that being the electron's position. Your system after the measurement will be consistent with the electron being in that position at the time of the measurement. In some sense, you retroactively change the gravitational effect of the electron by removing from your multi-particle wavefunction the parts where the electron pulled on the test mass and was measured in the end as being in a different position.
An easier toy problem
This is tough to give a really clear answer to because even if the electron is observed in one position, its wavefunction evolves with time (very quickly too!). So I can't just say something like "It's as if it was there the whole time"... no, because one nanosecond ago it could still have been in any position, and one nanosecond after the observation it will again be in a wide-ranging position-space superposition.
But consider as an alternative question an atom interferometer, where an atom is in a superposition where it travels down two radically different paths with equal probability separated by, in some cases, a whole meter. Let's say when the atoms are separated by a meter, we observe whether the atom is on the upper or lower path, and we find it is on the lower one. Then the moon's wavefunction (for example) is projected onto the subspace where it was gravitationally attracted to the atom following the lower path the whole time. It's as if the upper-path component of the wavefunction never existed (and never pulled gravitationally on anything).
However, as Andrew Steane points out, the moon's wave function will change by a very small amount upon this projection. If you subsequently did a measurement on the moon, you wouldn't be able to prove that this one atom had any gravitational effect until you repeat the measurement some insane number of times.
