even if the sd card is partitioned and even if they have the apps saved on the sd card, if the ROM they are using is not a2sd enabled then this wont work.
what happens with an a2sd enabled rom is that there is a "bridge" built between the partition on the sd card and the /system/app and /data/app files where the apps are normally stored.without this "bridge" the phone would not not to look for the apps in the sd card and will not see them
hope that answers your question.
oh and the nand backup has nothing to do with the restoration of the apps, again that is all do to the "bridge" created between the directories.