As you probably know, the head diameters of the two calibers are different. If you are talking about being competitive in registered group benchrest matches, you would probably be better off going another way. Like any factory action, a 40x would require a lot of work to be dimensionally correct enough to be competitive. This is not to say that it cannot be done, just that, given the cost, there are better ways to go, like a used rifle that is based on a custom action. While the .222 is a fine cartridge, (I have two, one of which is a competition rifle.) the 6PPC owns the sport, so to speak. It has the advantage of being able to be shot in all classes, while a .222 cannot be shot in sporter matches ( which require a caliber greater than .22). As far as using the action that you have, the bolt face can be opened up, and a SAKO extractor fitted, but the better option these days is to buy an aftermarket bolt, which can be ordered in a larger diameter to improve the fit, and the correct bolt face for a 6PPC. Even with the new bolt, there will be considerable work required. This brings us back to a used rifle, which would come with a usable stock, trigger, action, and the possibility of a barrel that has some life left in it, although I would not count on this.
