I'll note here that the Montanas had their design work essentially done by the time they were cancelled. USN design practice at the time broke the design work up into three stages. The first, Preliminary Design, was getting the design to the point of having a Spring Style ready to show people, one that showed them generally what the ship would look like, what it would have as combat systems, and how much it would cost; this is basically getting it to the stage of "getting SecNav and Congress to approve it." The second, Contract Design, took the work from Preliminary Design and did a much more elaborate version of it, one that was detailed enough that it could be used as the basis for competitive bids from the shipyards to get the construction contract (hence the name). The third phase, Detail Design, was done in concert with whatever yard was chosen as the lead yard for that class, and took the Contract Design and expanded it to the point where you had literally every single piece of the ship designed, to the point where you could just go ahead and start cutting metal as soon as you got the go-ahead from Congress and the Navy.
The Montanas were essentially through the Detail Design phase by the time they were cancelled; indeed, Illinois and Kentucky were originally going to be Montana-class ships, but were reordered as Iowas to get a couple more CV escorts out there. The only real reason that no actual construction work was started on the Montanas was because of a steel shortage that was later traced to misplaced priorities.
So no, Jersey wouldn't have to do any design work; I also say that they don't really qualify for the "napkinwaffen" status of the Ratte, the Monster, and the H-40 through H-44 designs, as we had them at the point where all we had to do was take the detailed design drawings to the molding loft, make the full-sized template copies of them, and then start cutting steel.
