I decided to have a quick look at some European Medicines Agency assessments for AZ and Pfizer vaccines along with Fluenz, the flu vaccine my kids got a couple of weeks ago. The EMA reports are long but are at least publicly available.
All three vaccines contain new ingredients never used before in pharmaceuticals and so get special attention. All three are genetically modified products with the AZ and flu vaccine being whole virus (the Pfizer vaccine is purely synthetic but uses modified mRNA)
The doubts about the COVID vaccines are have they been tested well enough and for long enough. Luckily, those reports have that information;
Astra Zeneca
Trials started in June 2020 with a data cutoff in December. The total number of subjects was 16,550
Pfizer
Trials started in June 2020 with a data cutoff of November. The total number of subjects was 37,796
Fluenz Tetra
Trials started in April 2009 and ran for 6 months. The total number of subjects was 5893
The trial periods were roughly the same length of time. Where things sped up, as others have said, was that there was an overlap between safety (phase I) efficacy (phase II) and clinical outcomes (phase III) trials with no pause in between.
Here are the reports if you would like to do your own research or have trouble sleeping;
Astra Zeneca -
https://www.ema.europa.eu/en/documen...-report_en.pdf
Pfizer -
https://www.ema.europa.eu/en/documen...-report_en.pdf
Fluenz Tetra -
https://www.ema.europa.eu/en/documen...-report_en.pdf