Adding ground to replacement sub panel
Since it is an existing installation you don't need the ground wire until you upgrade it.

Your inspector may even allow you to use the 3-wire method with the new panel until you replace the wiring.

Fuse panels are not necessarily obsolete until they are no longer serviceable.

You could use it as-is unless you have the need for more than 60 amps right away. If it is working fine now don't be in a big hurry to spend your money.

Good luck!