My son has a severe peanut and tree nut allergy. We have eaten at many quick service and table service locations at Disney World over the past 2 trips (2017 and 2018). So far all have been fantastic for our needs. They will show you the allergy menu and will take the time to explain things when asked.
I know you can now specify via mobile ordering about the allergy, but for us we would like to make sure we talk to someone about it first rather than rely on it going through a computer only. We don't mind waiting a few more minutes for additional peace of mind. Just a personal preference on that as I am sure it is likely safe, but as my sons allergy is severe we don't like to take chances.