This seems rather incredible that these two seemingly conflicting
announcements come on the same day.
The pre-print for the Nature paper by the BMW group was placed on arXiv in 2020 around the same time as the muon g-2 Theory Initiative paper (submitted on 8 Jun 2020 and last revised 13 Nov 2020, published here) that the Fermilab collaboration referenced in its announcement. The BMW pre-print was posted February 27, 2020, and last revised on August 18, 2020. (The 14-person BMW team is named after Budapest, Marseille and Wuppertal, the three European cities where most team members were originally based.) So, insiders weren't surprised.
Fermilab announced when its experimental results would be shared. Nature intentionally and strategically released the paper that had been in their offices awaiting publication for about eight months on the same day to maximize its impact. The theory presentation on Zoom this morning from Fermilab actually mentioned the BMW paper in passing although it didn't emphasize the point.
The BMW paper and the Theory Initiative paper use quite different methodologies to calculate the leading order Hadronic Vacuum Polarization (LO-HVP) contribution to the Standard Model muon g-2 calculation, which is the dominant source of theoretical uncertainty in the calculation. The BMW paper makes some refinements in the calculation that are very computation intensive (it took several hundred million "core hours" on seven sets of supercomputers to calculate) and is a straight up theory calculation (Quanta Magazine has a nice discussion of what this involved with a bit more depth). The Theory Initiative paper uses data on electron-positron annihilation products experimentally as a substitute for doing some key calculations that BMW does "on paper" (with BMW criticizing the Theory Initiative and others using similar methods for not transitioning the experimental data into the theoretical calculation in the manner that they think is correct).
Compared to the combined results used for the 4.2 sigma compared to the Theory Initiative paper, the BMW number is consistent with the combined experimental result at the 1.6 sigma level. The BMW paper claims 27% more uncertainty in its theoretical result than the Theory Initiative paper and the Theory Initiative result has been replicated (something much more difficult to do in the case of the much more computation resource intensive BMW calculation), which is part of the reasons that Fermilab decided to go with the Theory Initiative benchmark (another is that the Fermilab theory director is part of the Theory Initiative collaboration, while the BMW collaborators aren't affiliated with Fermilab).
A new pre-print making a more precise calculation of the hadronic light by light contribution to muon g-2 (reducing the relative error in that component of the calculation from 20% to 14%) was also strategically released on arXiv today, and would further reduce the experiment-SM prediction variant in the BMW prediction to 1.3 sigma (the two innovations do not overlap).
It will be interesting to see if the Lattice QCD/SM prediction theorists will be able to reach a consensus on this issue, which was a simmering behind the scenes controversy for months before this announcement. Multiple conferences on the muon g-2 calculation are planned for the next year (as the theory presentation from Fermilab this morning mentioned) and this disparity will be at the top of the agenda at all of them.
The stakes are pretty high. If the Theory Initiative is right, new physics beyond the Standard Model almost certainly exist and are right around the corner. If the BMW collaboration is correct, the prospect for new beyond the Standard Model High Energy Physics discoveries at experiments that are possible to conduct in our lifetimes is dismal.
An intermediate possibility is that the neither calculation secures consensus support, but that the QCD community concludes, based upon the disparity between the legitimately calculated Theory Initiative result and the legitimately calculated BMW collaboration result, that both papers have grossly underestimated the amount of theoretical uncertainty present in their SM prediction calculations. This would leave us in the same limbo we've been in on muon g-2 for the last two decades.