My apologies for overlooking your original request for details on annealing.
I would recommend calling for a dry hydrogen anneal (as mentioned by EdStainless) at 1600 degF for a minimum of one hour at temperature.
One thing to keep in mind, the annealing can distort the part somewhat, enough to ruin any tight tolerances. I'd suggest having the annealing done before machining any tight tolerance features onto the part.
The machining afterwards will only degrade (not destroy) the magnetism in the regions where the machining was done. The bulk material will remain in the annealed condition. Hopefully the only machining necessary after annealing will be tapping a few holes or other minor features.
I read your post. This is the first time that I have used eng-tips. My company has been in business for 20+ years selling Low carbon (c1002) strip & Rod for electromagnetic applications. I am sure I can help you! We stock a variety of sizes .250 - 6.0" dia.