I can only speak for my surgery.
No, we do not get referred to a podiatrist automatically. The nurse does our foot checks for sensation and pulse. However, there is a self referral service to a podiatrist for people with problems, which is free. There is a waiting list, and once she has examined your feet you are discharged unless something needs attention, in which case she will see you again until it is resolved. Other than that, it is a private one.