Look at it this way: whatever size a phase 2 trial would have been it would have delayed the deployment and that would have cost 100's of thousands, quite possibly millions more dead and many tens of millions if not hundreds of millions more people infected. The chance that something major would not surface in the vaccinated population so far that would have surfaced in a phase 2 trial is nil, because the number of people that have now been vaccinated is orders of magnitude larger than the largest ever phase 2 trial.
You do a phase 2 trial when:
(1) you have serious concerns about possible side effects
(2) you have serious concerns about efficiacy
(3) there is no pandemic going on
(4) you want to market your product
The people that made the call to skip the phase 2 trial were experts in their field and apparently decided that (1) was a possibility so they decided to monitor the first couple of million vaccinations closely to see if there would be side effects and whenever those occurred they halted vaccinations until they found out what was up and whether or not there was a way to proceed or to call the whole thing off. AZ in particular ended up being banned for several applications due to an abundance of caution leading to the destruction of a large number of otherwise viable vaccines (and some got donated to countries with less stringent criteria).
(2) never was in doubt, though the exact numbers would only come out after a while and some of the longer term effects in terms of protection and protection against mutations we are only finding out now because some time had to pass for those effects to become apparent.
(3) is obvious, there was a pandemic going on and so every day counted because an exponential process only yields to one thing: fast and mostly accurate decision making, slow and perfect decision making gets you killed every time.
(4) was not much of a problem, regulators and governments alike were begging the companies producing the vaccine to get on with it, because they realized that they had botched their strategy so completely that their only hope was to have a vaccine developed and shipped as fast as possible. Under normal circumstances regulators would have been slower to ensure that all the i's were dotted and all the t's were crossed but in this case they were more than willing to take the -small- risk that a phase 2 trial would have uncovered something really bad that would not come out in the first batch of vaccinations to the general public.
Note that before the general roll-out there had already been a substantial number of people that volunteered to receive the various vaccines and they more than anybody else deserve a lot of credit because they allowed the pharmaceutical companies to compile a mostly complete list of side effects and things to watch for.