OP - my nephew also has a nut allergy and has been cruising since he was a year old (he's 8 now). Watching his parents navigate this has been an education but it has never stopped them from cruising. They do contact the cruise line in advance (prior posts have listed the forms and numbers you can call) and worked with food service and cruise staff on ship. The 1st thing my sister says to our waiter/waitress is that her son has a nut allergy. I have yet to see an issue though , full disclosure, this will be their 1st HAL cruise.
From what I have read, HAL is very conscious of the health needs of those sailing with them so odds are there shouldn't be an issue as long as you communicate.