Are there any non-surgical procedures to pin back my ears? (Photos)

No, there is no effective nonsurgical modality to  set back prominent yours.  A well executed otoplasty procedure will likely be in your best interests.  Seek consultation with board-certified plastic surgeons who can demonstrate lots of experience achieving the types of outcomes you would be pleased with. You may find the attached link helpful to learn more. Best wishes.

There is a new technique of Earfold that is less invasive than traditional surgery.  This involves a cm incision on front of the ear and placement of a permanent clip that folds back the cartilage.  The cost of this would be about £3000-3500.  

There is now a minimally invasive procedure to pin protruding ears. It is the Merck stitch method. This comes very close to meeting your wish of having a non-surgical procedure. With this method, ears are not cut open anymore. The only wounds are stitch points and 2 to 3 small, roughly 2-3 mm long, so-called stab incisions on the back of the ear. A bandage must therefore not be worn after the operation. One may check and co-determine the result in a hand mirror.