Copyright issues like this revolve around licencing arrangements. The problem you have here, is that there is no such arrangement in place. You only "obtained permission" to use the logo, but permission in itself does not give you any "exclusive right" of use. Exclusive rights are the core of copyright law and these rights can only be assigned to another if there is some kind of licencing arrangement in place between the copyright holder and that other person.
So in relation to someone else using the logo, based on the info you've provided, I would say that it is up to the school to determine if any breach of copyright has occurred. In short, if this other manufacturer has not obtained permission like you did, then their use of the logo is highly likely to be in breach of copyright, but it is up to the rights owner to take any enforment action.
You however, can take action in relation to any arkwork that you designed yourself, provided there is no contractual arrangement that assigns the rights of that artwork to someone else. The good news here, is that there is no such contract in place here either...
We have permission of course
This indicates that there is no licencing arrangement in place in relation to the logo - which means that the school retains exclusive rights. Without a formal licencing arrangement, your use is "by permission, but not exclusive".
No
contract with school but advertised as provider.
This indicates the same, but also that there is no contract in relation to your own designs/artwork, therefore you retain the rights to your own work.
Although I am not a lawyer, I do have qualifications in the music industry and have studied copyright law. The basics of licencing are pretty much the same across the board, whether it be literature, artwork, music, etc.