So, problem is from another computer in the network to server?
Check firewall if port 3306 is open. You also have to allow (in MySQL) access from network for username that is connecting.
I suggest you try connection with HeidiSQL
https://www.heidisql.com/, and when you can connect with HeidiSQL then try your application.
In application check if you have correct HostName (IP or network address), port, username, password, maybe your antivirus is blocking your app. Check bitness of your programs: if one is 32-bit and other is 64-bit same libmysql.dll won't work.
Try to fill LibraryLocation to point to libmysql.dll.
You have to systematically check one by one issue.
And you'll have to give us more details if this won't help you.
What components are you using for mySQL connection? In ZEOS you have to set correct protocol.
What OS-es are in each computer?