I would remove all metal parts from the guitar. You have pictures to see where and how everything fits together, I would next put all in an old plastic container and spray it all with oil like WD-40. Next I would take all apart, all screws out and so on. Now I would all bath in machine oil (like available for sewing machines), take it out, put all on paper towls and dry them. When in the oil you also could use an old toothbrush to remove hard dirt. I would not use any metal polish as that looks not so good on vintage guitars. Just use some old cloth like from a T-Shirt at the end and rub everything dry and assemble and reinstall.
EDIT: I just see that you probably can't remove the 6 screws and saddle pieces from the bridge so just when in the oil do turn the screws left and right to move the saddle pieces all around and thereby cleaning the screws.
Ralf