An excellent article that explains why the fact that a COVID-19 virus has been developed, tested and (in the UK) approved in a matter of months is not, in and of itself, a cause for concern.
I suppose that I should state the obvious: life is not without risks. There is no guarantee that any approved vaccine will not harm anybody. What we do know and should take to heart is that the COVID vaccines which are ultimately approved by major regulators have been studied and deemed to be sufficiently safe for use by the population as a whole.
Also, anyone who refuses to take any risks at all when it comes to vaccines should ask themselves why they routinely take risks in their day-to-day lives: crossing at a traffic light, flying, riding in or driving an automobile, eating a meal which (for a variety of reasons) might contain dangerous bacteria or other substances, getting out of bed in the morning (obviously, the risk of falling and breaking your neck is much lower if you just stay in bed), etc, etc, etc.
P.S. I fully intend to be vaccinated for COVID-19 when my cohort becomes eligible, and getting vaccinated becomes feasible and relatively hassle-free. My current best if somewhat wild guess is that this will happen sometime in the spring or summer of 2021.