This will require a new launcher. The VLS systems on the Zumwalts are too small, even the Mk 57, for the diameter of the new missiles. While entirely doable, I'd be a little surprised if the launchers can be designed, bought and installed on the Zumwalts in that time frame.
If they use the same system the USN is planning for the Virginia and Ohio classes, it will be something of a waste of volume and only have between 14 to 21 hypersonic missiles. They really need to get a tightly packed VLS like the Mk 41 (but larger). The number of cells possible would mean between 32 to 48. If they did the cell design right, they could pack in multiple SM-x, ESSM and others in subdividing cells. They did it for the standard MK 41 for 4 ESSM/cell.
However, I am pretty confident the USN will find some way to screw this up. They've done a LOT of really dumb things in the last couple decades.