Actually, it's not as cut and dried as you might think.
Can a medical doctor perform surgery on an animal? Absolutely not. The role of a vet is covered by statute law, and it is only legal for someone holding a veterinary degree to perform operations and procedures on an animal as covered in the relevant Acts of Parliament.
However, as to your question as to whether a vet can perform procedures on a human, the answer is YES! Strangely enough, the role of a doctor is overseen by their own professional body, the GMC, and not necessarily mandated by statute law, and, in effect, the GMC is essentially just an insurance guarantor should a claim be made against a doctor for malpractice etc. In fact, anybody can perform procedures on anybody else, as long as they have consent etc. To illustrate the point, many HCA's carry out clinical work that most junior doctor's might not even be competent at in their F2 training post, and yet the HCA doesn't need any qualification to do or insurance cover should something go amiss. It's a strange world.....