We sailed on Symphony with my daughter, who has anaphylactic reactions to tree nuts. You can read about our experience in my trip report (in my signature) but overall it was great. I sent an email in advance, and we chatted about their procedure. When we boarded we ate at a specialty restaurant and mentioned DD's allergy.
At dinner in the MDR we mentioned the allergy to our team, and were connected with the manager? head server? I don't remember, but he was great and he checked on us every night. He also put together a special dessert for us, which actually ended up having one of DD's other allergies. He was so focused on the tree nuts he forgot about the coconut, so you still have to be super aware of everything that is served, but that applies wherever you are.
There was bread in the basket that contained tree nuts (WHYYYYYYYY???????) so after the first night they gave DD a plate with nut-free bread.
Aside from the one dessert faux-pas (which again, was a dessert they don't usually serve in the MDR so the ingredients were not as familiar to our serving team) DD had dessert every night. Actually she had dessert that night, just not the one that had coconut.
We avoided the buffet on this cruise, but we have dined at buffets on other cruise lines and it was fine. DD prefers things that are normally nut-free like fettucine alfredo or burgers and fries, so buffets are not a huge deal for us.
You need to be aware, and be proactive, just like you would at any restaurant or resort. But Royal will be able to handle the allergy. They are very aware and there are hundreds of people with allergies sailing every day. They will do their best!