Pfizer's was approved last week and is in distribution. Moderna's was approved yesterday, so distribution is just beginning.
AstraZeneca's would have been on the same timing track, except a manufacturing glitch resulted in a substantial number of people getting a half dose for one of their two doses. Those people did well (as a group, better than average, but they were mostly people 55 and younger, IIRC), but the dosing inconsistency screws up the efficacy calculations. My guess is that AstraZeneca's vaccine may receive approval in January.
The Johnson & Johnson (Jannsen Research) vaccine's Phase III trial reached full enrollment (44,000) a couple of days ago, so it could receive approval as early as March. Novavax's vaccine probably is in a similar stage and likely timing.
As to the DVA priorities thing, I think it is probably a symbolic PR-pander that will have little or no practical effect.