I would suggest, if you are keen on postgraduate studies, to find the course you would apply to in the future and check their entry requirements. There, the universities list the subjects and often say you'd need at least a 2:1 in your BA/Bsc.
I would guess(!) that biochemistry would be a better choice because Applied Medical Sciences (depending on the universities) focus primarily on humans. Biochemistry is often studied in the latter as well, plus, it gives you the basics that (an educated guess here) you would need for vet med. I'd suggest you compare the modules that you would have studied in the veterinary medicine degree and compare those to biochemistry and applied medical sciences. Probably best then to pick the one that overlaps the most (which will probably be biochem).
The entrance requirements for graduate entry to vet med at the a) Univeristy of Edinburgh are a 2:1 in a biological/animal science. "Subjects studied MUST include Biology/Zoology, Physics, Biochemistry, Organic and Inorganic Chemistry and Mathematics/Statistics.
b) Bristol accepts "Applied Anatomy; Biochemistry; Biology; Biomedical Science; Cancer Biology and Immunology; Cellular and Molecular Medicine; Chemistry; Medical Microbiology; Neuroscience; Pharmacology; Physics; Physiological Science; Veterinary Nursing and Bioveterinary Science; Veterinary Nursing and Companion Animal Behaviour (from 2022); Virology and Immunology; and Zoology".
Seems as if there are only three graduate entry programs in the UK, so you might have to apply for a second undergraduate degree.
If you do the steps I've outlined above, you should get a clearer idea which option works best for you.
Let me know what you have decided to do at the end.