To start with, your mother is over reacting to nothing. The only parts of Japan that have any problems are the North East of Honshu. Tokyo and Chiba had some damage from the quake, but are in no danger at all from any radiation.
You shouldn't have any issues connecting through Narita. If you get a good check in agent at ICN, you should also be able to check through across the two tickets.

If you still go ahead and change the routing, you'd have to add NRT-ICN as a surface sector, which puts you over the sector limit and would require dropping a later flight.
You won't be able to do ICN-HKG-ORD. HKG-ORD service doesn't start until September.

If you must avoid NRT, I'd get yourself to HKG, reticket with a NRT-HKG surface sector, swap ORD/JFK or SFO/JFK around and drop SCL.

AKL-SYD-BKK-NRT/HKG-JFK-CUN-ORD-SFO-LIM-IGR-EZE-MAD-IST-LHR-HKG-AKL
AKL-SYD-BKK-NRT/HKG-SFO-ORD-CUN-JFK-LIM-IGR-EZE-MAD-IST-LHR-HKG-AKL