Cambridge student here! (I haven't been on this site in years 😂😂).
NatSci - very broad course, you do three science options in first year and maths, then specialise later. For biochem, you'd be looking at biology of cells and chemistry in first year (with another science, could be a biological or one of the physical options), then biochemistry and moleculr biology, one of the chemistries, and something else in second year, and then from third year onwards biochem all the way through.
You're not locked in to anything, and you can always switch. Lots of lab time (up to 12 hours first year, up to 18 hours second year, lots of variability in third depending on options). Downsides: Saturday lectures in both first and second years if you're taking Chemistry options.
Oxford - much less freedom, much less choice, but if you don't want to have to study many subjects, possibly a better option.
The admissions test (NSAA) is fairly straightforward - it was a lot of multiple choice when I did it, not sure if it's changed since! Don't choose based off the admissions test (unless you're avoiding STEP, because that paper is evil).
Feel free to ask any questions - I'm a phys natsci, but I did a bio option in my first year and I have friends who do biological or chemical options.
I can't promise I'll answer (I'm just here to procrastinate before I get back to revision), but if I'm online again, I will.