Cobby started singing while attending Ealing Academy, before becoming a chorister at Worcester College, Oxford and, later, the City of Oxford High School for Boys, where he won a competition to perform at St Paul's Cathedral. It was during this time that Cobby's love for acting grew, and he visited the Oxford Playhouse every week. One day he noticed an advertisement for a job for a young boy or girl at the Playhouse; this turned out to be sweeping the stage, but the manager encouraged him to continue his studies and later return to acting.