Moving to new servers was no problem.
Moving to a new server, and moving to a new domain name, are totally different things.
Moving to a new server should have no impact at all.
Moving to a new domain name requires a whole re-indexing , a whole re-calculation of the ranking stuff.
At the end, the new domain, should have the same performances as the old one. (Not better, not worse).
A few years ago, I would have said, that 2 weeks was the time for Google to catch up the change. Today, I would say, at least a month.
It's better to do this during a period where the business has the less activity.
To make the transition smoother, but a lot longer, is to keep the content on both domain names , and use the canonical tag to select the new domain. Then, when you observe that all the traffic is directly going to the new domain, you can set up the 301 redirects. However, it can take months, and might not be convenient to keep up the content on both domains, depending of your infrastructure.