Numerical results of precision tests in perturbative QFT

As a general source, you could look at the CODATA22 review:

Mohr, Newell, Taylor, Tiesinga: CODATA Recommended Values of the Fundamental Physical Constants: 2022

Below, I try to give some more specific references. Each of them contains some breakdown of theoretical contributions and comparison with experiments. But note that $-$ as the other answer correctly remarks $-$ the numerical values of separate diagrams are in general meaningless as they are not gauge invariant. This is not a problem in lowest order (e.g. the one-loop energy shift arises from bound-electron self-energy and vacuum polarization, both contributions being independently gauge invariant), but in higher orders, the best you can get are gauge invariant subsets of diagrams.

Hydrogen-like ions

• The general nrQED theory of predominantly non-relativistic (low-$Z$) hydrogen-like ions and muonic hydrogen (Lamb-shift, hyperfine structure, etc.) is reviewed in

Eides, Grotch, Shelyuto: Theory of Light Hydrogenlike Atoms (2000)

• A less detailed, but more up-to-date review of the Lamb shift:

Yerokhin, Pachucki, Patkos: Theory of the Lamb shift in hydrogen and light hydrogen-like ions (2019)

• Also Lamb shift, but for systems with stronger relativistic effects (higher $Z$):

Yerokhin, Shabaev: Lamb shift of $n=1$ and $n=2$ states of hydrogenlike atoms, $1\leq Z\leq110$ (2015)

• An old but very good paper (the numbers might be somewhat outdated, but the discussion is very detailed):

Erickson, Yennie: Radiative level shifts, I. Formulation and lowest order Lamb shift (1965)

Positronium

Probably the most recent review on positronium is

Adkins, Cassidy, Perez-Rios: Precision spectroscopy of positronium: Testing bound-state QED theory and the search for physics beyond the Standard Model (2022)

It compares computed and experimental hyperfine splittings and lifetimes of Ps (note that hyperfine splitting is a rather unfortunate name for Ps; it just means the ${^3S}-{^1S}$ energy difference in this context).

Free-electron $g$-factor

A recent review about $g_\text{e}=2(1+\kappa_\text{e})$ (where $\kappa_\text{e}=\alpha/(2\pi)+...$) is

Aoyama, Kinoshita, Nio: Theory of the Anomalous Magnetic Moment of the Electron (2019)

There are similar, but more controversial computations/measurements concerning the free-muon $g$-factor, but I do not know enough about that; hopefully someone else can explain it.

Bound-electron $g$-factor of hydrogen-like ions

The $g$-factor of the electron gets corrections from the binding to the nucleus, which can lead to significant deviations from the free-electron value: $$ g_\text{e,bound}=\frac{2}{3}\left[1+2\sqrt{1-(Z\alpha)^2}\right]+2\kappa_\text{e}\left[1+\frac{1}{6}(Z\alpha)^2+...\right]+... \ , $$ further corrections involving higher QED, nuclear recoil and nuclear structure corrections, and so on (as usual). A breakdown of the different contributions for Si$^{13+}$ and comparison with experiment can be found in

Sturm, Wagner, Schabinger, Zatorski, Harman, Quint, Werth, Keitel, Blaum: $g$ Factor of Hydrogenlike $^{28}$Si$^{13+}$ (2011)

Helium

• Fine-structure splitting of the $^3P$ states of He (proposed as a way to refine the value of $\alpha$):

Pachucki, Yerokhin: Fine structure of helium-like ions and determination of the fine structure constant (2010)

• Static dipole polarizability of He:

Puchalski, Szalewicz, Lesiuk, Jeziorski: QED calculation of the dipole polarizability of helium atom (2020)

...and so on

This list could be continued infinitely (with e.g. the high-resolution rovibrational spectroscopy of H$_2^+$ and H$_2$), but these are the precision tests of QED that I can assign references to off the top of my head.

Frame challenge: Individual Feynman diagrams are not gauge invariant, so the value of a single diagram is not meaningful. (At least if you have gauge bosons around, as in the $g$-factors you ask about.)

Gauge-invariance is guaranteed for the full amplitude by the Ward identity, $k^\mu \mathcal{M}_\mu = 0$, where $k^\mu$ is the momentum of a photon leg, and $\mathcal{M}_\mu$ is the amplitude with the polarization vector stripped from it. This ensures that under shifts of the photon polarization vector $\epsilon^{\mu} \rightarrow \epsilon^\mu + C k^\mu$, which is the on-shell gauge freedom, the amplitude $\epsilon^\mu \mathcal{M}_\mu$ does not change.

However, if you split the amplitude into contributions from multiple diagrams $\mathcal{M} = \mathcal{M}_1 + \dots$, then each individual diagram will generically be modified by such a gauge transformation.