This is just from personal observation and I've probably missed quite a bit of nuance, but AFAIK all the meals for a service are pre-loaded in the ovens and heating starts during climbout. There are plenty of ovens. You can often hear the whine as the ovens start up. Modern ovens have a menu where the FAs specify what is to be heated (bread, meat etc...) and then they press a button. The display then shows a countdown.
The biggest trick to a smooth running kitchen (aircraft or domestic) is timing. Prep things in the correct order so that they will be finished at the same time. However I don't think that is quite as important in economy as the meals will have to stay warm for quite a bit. For items like bread, however, you can only really heat them once. Keeping them warm will eventually lead to hardening.
It runs smoothly due to lots of practice.