Don't worry too much about not having a mandrel, they are easy to make. Not tapered mandrels (those are trickier), but you don't need one of those. This is what I would do.
Cut your rough stock to size (do an extra or two, just in case you make a mistake on one, it's always a good idea when you are doing multiples of a part. Then run each rough blank through each of the setups below. I recommend running all the pieces through each setup in a batch.
Setup 1
1) Chuck a piece up
2) Face it
3) Drill
4) bore to the smaller size hole (Your 5/8" diam)
5) Bore to your larger diameter hole
Setup 2
1) Chuck up one of your partially finished parts with the faced side facing the chuck
2) face to length
Setup 3: Interlude (only needed once, do not remove from lathe)
1) Make up a mandrel by chucking up a piece of 1" stock. Make sure to center drill and use a tail stock center
2) Turn part of it down to 0.867
3) Turn the end down to 5/8, and thread all but about 0.5" of that.
Setup 3
1) Load one of your parts on, and secure with a 5/8 nut.
2) Turn the outside to size
3) Cut the groove
You could also try using superglue to secure the part rather than a nut, but you will have to use heat to release. If you don't remove the mandrel from the chuck, everything will stay very concentric. If you want to be able to remove the mandrel and keep repeatability, I would recommend making the mandrel by turning between centers.
