The "10-15 year" process you keep referring to has more to do with economics and available case numbers than testing protocols. Much of the delay in traditional vaccine development was due to securing additional funding between trial stages and setting up the infrastructure and processes to manufacture the vaccine once it was approved. Governments throughout the world have provided guaranteed funding so now the pharamceutical companies can work on multiple aspects of the vaccine at once, rather than the step-wise approach that needed new funding with each bit of progress.
The other reason why this process can be accelerated is that with a disease with as high an incidence as COVID-19 (in the millions currently), the "n" number for the studies gives them much more power and quicker turn-over than for a disease with lower incidence, like for example, measles, who's annual incidence before vaccination was only in the 10s of thousands.
As for safety, no vaccine has ever failed a trial or needed to be withdrawn because of delayed side effects. Most of the side effects of vaccines appear within days of receiving the dose. The most severe (and fortunately, extremely rare) complication, Guillan-Barre syndrome, arises within days to weeks after the causitive event, not months to years later.
I would think the reasons for the accelerated development during a pandemic would be obvious, given how this virus has disrupted the lives of virtually everyone on the planet. Vaccine development was traditionally slower not due to policy, but due to a lack of urgency.