What is the minimum downtime way to switch server data from bad hard drive to new hard drive in a remote server hosted in another country?
Server has only one physical HDD, no raid. Around 200Gb of data and MySQLs that are per my knowledge not recommended to backup as files, but mysqldump them.

There is no way for anyone to answer this. This will depend on how much bandwidth you have, and what steps you've taken already. If you already have verified backups of your data, then copy them to the new server, test them, and bring up your database. After that, you'll have to do whatever you have to, to make that server function as the old one did. Either you have to move your DNS record, change the host name/address, or modify any one of MANY different services.

You don't provide details, so we can't guess. It could take anywhere from ten minutes to ten hours, or longer.