(OP here)
If anyone cares about how this played out, I got an interview invitation at Imperial pretty soon after sending off my UCAS application. My interview experience was overwhelmingly opposite of that at Cambridge. My interviewer (a researcher/lecturer) was extremely kind - I was asked to think of, and talk through an algorithm to solve a problem based on calculating the area formed by rainfall across a hilly ground. It was presented on a kind of histogram. It was hard to formulate the solution at first but the interviewer gave very helpful hints (without giving away the answer) which led me to answering well enough. Other than that, I was asked about the projects listed in my personal statement, as well as my post-degree plans.
The Cambridge interview on the other hand was on a miserable December morning in a cold room somewhere in Churchill College, with 2 very unnerving professors almost mocking me as I cackle in front of a logic gate problem involving streams and pipelines of water... "exactly... you can't make water out of nothing ._." was the exact quote right before I was told my time was up. (The problem was of moderate difficulty, unless your heart is doing 120 BPM). By the second interview my confidence dropped to the point where I couldn't sketch a simple (sinx)^2 curve to save my life. I was, however, able to tackle the rest of the 7 maths problem selection given to me. I wasn't asked about my personal statement or anything other than the problems presented. Eventually I stumbled to the exam hall where I took a maths test lasting an hour, in the style of a UK maths challenge paper (but harder). Wasn't surprised to see a rejection letter, but hoped that Imperial would work out.
Hope was restored when I eventually received an e-mail:
On the strength of your application, together with your interview performance, we are pleased to inform you that you will shortly be receiving a conditional offer for the course G401 at Imperial College London.
Offer:
A* in Maths
A in Further Maths
A in Computing
A in Economics
A in Extended Project
Grade 2 in STEP 1 or Merit in Advanced Extension Award (AEA) Mathematics Paper
I met my offer and am excited for freshers week starting next Saturday
It was hard work and months of very nervous waiting, but it paid off.