I used to live a hour away from St Augustine and have visited there many times. I cannot however recommend any resturants outside a major chain one. Smaller resturants are just not as equipped to handle allergens and most dont have official procedures in place. It can quite simply be a nightmare. Especially in the loud popular touristy places.
Its not exactly fancy food. ( understatement ) but Cici's pizza is located in St Augustine and you can eat absolutely anything from their pizza bar and not have to worry. It states right on their website NONE of their food including desserts contain nuts or are made in a facility with nuts. Its not exactly 5 star dining but it is a safe bet if you need to grab something quick.
Here is the address
115 Jenkins St
St. Augustine, FL 32086
Is it a adult or child with the allergies ? If its a child I was going to say you could bring something for the child and adults could eat out somewhere else. There are several good resturants in the area I could recommend in that situation. Most kids dont mind geting a BK kids meal and I do believe they are also nut free as well. I cant see a resturant saying anything about you brining something in. Unless they wanted to be responsible for any issues and most do not. ( Have to double check on the BK site about the new funnel cake sticks though)