Cambridge maths, at least those not applying to Trinity, gives interviews to most (perhaps all competitive) home applicants. My knowledge may be a few years out of date, but every pre-interview rejection I knew of for a while was either from Trinity or to an international applicant, or both. Might be all both, I'm afraid my memory is failing me a bit here. Trinity gets twice the number of applications of the next most popular colleges, and at least five times as many as most other colleges, so presumably they have no time to interview everyone. There is no pre-interview pooling so if a college does not interview you then your application is dead right there.
Whether you get the offer will then mostly be on the interview and any at-interview assessments, and then whether you get in will be largely based on your STEP mark and the details of your script - perhaps looking back at interview scores in borderline cases. It is rare for someone to get the STEP part of their offer and miss the A-level part.
This is all to say if you're a home applicant applying to not-Trinity, you can expect an interview. If you are an A*A/A*A* student in maths/FM, you're in a position to consider Oxbridge. It's only one application and there are "safer" options you can put down. I will quote a Queens' admissions representative verbatim that they do not care about GCSEs. A good personal statement should get the interview at a good start but I don't know if it'd offset poor interview performance and especially poor STEP.