if you are using cPanel on both servers and you use the transfer feature provided in cPanel with root access to both servers, then about an account using 5gb of space would take maybe 5-15 minutes to move.
doing it manually by FTP (downloading to pc and reuploading) would increase the move time 5-10 times depending on the person's internet speed.
but again, it depends on the size of the site/account and the method being used. the numbers I mentioned mean pretty much nothing in most cases.
5 GB in 5~15 minutes if you own both servers, but if you moving between providers sometime it takes hours if not days to move an account > 1 GB from an overloaded server with other providers, there are many factors to consider like how well both servers are connected and the load on the servers at both end.
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It all depends on multiple variables such as server speed, network speed, size of data, number of files, etc. Without knowing any of that, you can't really determine how long transferring data will take.
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Is it just data or are they also transferring MySQL databases as well? If that's the case, I'm assuming the company would also check the configuration files for your scripts to ensure the proper directories are being used if required, as well as updating the MySQL information if that has changed from your previous host.
Assuming the server you're transferring from isn't extremely slow, it'd probably take about an hour or so at the most. It'd also probably be easier if you were transferring to the same control panel, company, etc. Just curious, how long has your company taken?