So I can leave and re-enter the US on my UK passport on the VWP (Visa Waiver Program)? Is the I-20 still required? SEVIS?
The I-20-M-N is required only for aliens seeking admission to the United States under an M-class visa. It is not therefore required of or available to aliens entering under the VWP.
My attendance at the school will be to finish the minimum hours required and at the same time, fine tuning the manuevers, with the CFI, in preparation for the Check Ride.
In the broadest sense you are proposing to embark on a course of study. Aliens are required to obtain the appropriate student visa if entering the US with the intention of pursuing any course of study. This point is addressed in a notice from one of the US government departments in a document linked to by a moderator some half a dozen pages back in this thread. In short it is not lawful, in terms of my personal interpretation of both the INA and the aforementioned notice (referencing an amendment to the CFR), to enter the US under the VWP to indulge in a course of study. SEVIS registration is required of persons entering under M-class (non-academic) and F-class (academic) visas.
What constitutes "finished studying"? I have taken the FAA Commerical written exam already. The only studying I would be doing is "home study" in preparation for the Check Ride. I'm sure that the CFI will have a "ground" session too before signing me off for the Check Ride.
In my opinion (the relevant federal agency may very well adopt a different one) this (and your remark I have quoted above) constitutes being instructed and thus because a course of study is sought you ought to be using an M-class visa.