I am also not a native English speaker, so I know that one can be more fluent in their native language than in learned languages. Even though, I understand the pressure that public speaking has on us stutters regardless of the language. The course also contains a part where you'll practice public speaking, watch some youtube videos of the program and you'll get an idea of what to expect. Since you're in the UK, there's a course coming up early in February in London. Make sure to let us know how it was if you decided to go.