Is this a networked hard disk or a simple portable hard disk that you connect to the computer you're working on? If the latter, simply save the catalog file on the external disk too and use that catalog file on both machines. This will make all your edits transfer.
You need to put your photos AND LR catalog file (.lrcat) on the external drive, and make sure the external drive has the same device name or drive letter on both computers.
I have been using LR this way for several years and it works fine as long as you MAKE A BACKUP of the external disk somewhere, in case you lose or drop/damage the external drive which is easy to do. Nightly, or sometimes more than once a day, I copy all changes from my working external disk to another similar-sized external disk. I have lost a disk, once, and dropped/destroyed a disk, once, but recovery is as easy as using what used to be the backup disk as the primary one, and buying a new disk for the backup device. It does take some hours to copy the 1TB of photos to the new backup disk, but once that is done, it usually only takes a few 10s of minutes or less to copy what is new or changed each night.
This was very helpful. Many thanks