The only way to become a psychiatrist is to do a medical degree, the usual two years as a Foundation doctor, and then psychiatry specialty training.
An undergraduate degree in psychology does not provide any shortcuts and would add nothing to your application. It's an interesting subject, but most of the material you study at undergrad level will not be clinically relevant. Graduate entry medicine courses exist (I'm on one) but these are even more competitive than standard entry medicine and they require higher scores on the admissions tests. If you want to do medicine, the best and simplest way is to retake A-levels and apply to courses that allow resits.
Remember that psychiatry is a tiny portion of an incredibly broad degree, so you need to be confident that you would enjoy that breadth. If you're not keen on that or you don't think you could realistically get the grades in the required subjects, have you looked at things like mental health nursing, social work, or occupational therapy? These professions enable you to go on to do postgraduate training in a psychological therapy if that's what you're interested in. I've had several therapist colleagues who were RMNs, and who got salaried CBT training posts on the strength of that previous experience.