Pursuing physics/engineering through the natural sciences?

Currently, I'm taking three sciences- and am really interested in the natural science course at Birmingham due to the amount of flexibility it gives. As I'm not taking maths, I know that a pure engineering/physics degree is out of the question, but is it possible to pursue engineering /physics purely through the natural science course? On the website, it doesn't specify as to whether or not maths is needed for the engineering course; however Is it at all possible that they let me onto the physics course (within natural sciences) without having A- level maths? Any help is appreciated.

Highly doubtful.

Maths is required for a reason - they expect you to know the content of A-level Mathematics in order to do the content of the physics and engineering courses. It's not an arbitrary requirement - first year physics (and to a lesser extent, engineering) consists of "re-learning" A-level Physics, with calculus - which is taught in A-level Mathematics. They concurrently teach more maths, including several major topics from Further Mathematics and developing calculus from the single-variable calculus seen in A-level to the multivariable and vector calculus required to do Physics at the university level, as well as applying it to new topics in differential equations that are the de-facto tool to model physical systems.

You can contact the university but realistically they won't allow it, and if they do I would question the wisdom, as having personally having been allowed onto a course for which I did not meet the prerequisites, it's a huge struggle and rarely works out well.

If your interest is in Physics or Engineering then I would advise you apply to such a course with a foundation year, at Birmingham or elsewhere. Alternately, you could take A-level Maths (and AS/A-level Further Maths, if desired) in a gap year and apply hence with the correct qualifications (this would also allow you to apply to Cambridge NatSci, which does allow you to pursue Physics or Chemical Engineering through the NatSci tripos.