After some careful research, the designer of Kate's coat has been identified! The velvet-collared coat is by London based designer, Katherine Hooker. This coat was previously worn by Kate in 2006 at the Cheltenham Gold Cup horse race (see picture below).
According to the Telegraph, the mid-calf length coat was shortened by Katherin Hooker herself so that it would hit just above the knee. The coat is known as the "London Contrast" coat and is made from Shetland herringbone with a brown velvet trim. Prices range from £520 (Express Collection Fabric) to £1,175 (Special Collection Fabric).