There are no legal requirement that have to be met when offering this type of training.
Accreditation simply means that the training company has paid an insurance company to provide accreditation in order for their students to purchase insurance from the insurance company upon completion of the course. Students may find that they cannot purchase insurance elsewhere and so might be tied to one insurance company.
You’d need to check with ABT whether they require the person teaching the course to be accredited with them directly.
Accreditation in this way does not guarantee any minimum level of quality or competence so you should always carefully check any online recommendations before booking the course. These types of courses are not equivalent to a longer course such as an NVQ/VRQ in Nail Technology etc. but tend to focus on a specific sub topic such as Gel Nails. However, depending on the trainer, they can sometimes provide more in-depth training on that particular topic and be good value for money.
With most training courses, your only redress through the courts is from breach of contract/misrepresentation.