This seems like a case where the precautionary principle is misleading. Maybe there's a small chance that the vaccine can be harmful for a few people. Meanwhile there's a large chance that not having the vaccine can increase your chance of getting COVID. Doesn't even a week's delay in vaccinating predictably increase the number of deaths in a country, perhaps by a large amount given the nonlinear dynamics of epidemic…
One of the many unknowns about these vaccines is the length of time they give immunity, this can only be determined with confidence by looking at the data after a sufficient amount of time has passed. If the immunity only lasts for say 10-20 weeks, then getting it one week early, would mean the immunity goes away one week early as well. So in this case, I'm not sure there would be a major measurable impact. If the immunity lasts for say 50 weeks, that would be a different story.