Fastest way to change a 600GB table indexed key datatype from INT to BIGINT
Assuming your table does not have any triggers on it, you should consider using pt-online-schema-change since it will allow you to ALTER the table without locking it.

It will still take a fair amount of time given the size of the table.

Also, with this method or with ALTER TABLE you will need to make sure you have 600 GB of extra disk space to support two copies of the table while it is being rebuilt.