But you have it backwards. When programming is introduced as maths people see how maths is more than ‘ arithmetic, polynomials, integrals, and derivatives’.
Maybe it's just semantics, but I think our common perception of math is too narrow.
Now we have discrete Computer Science at GCSE and A-Level.
[1] https://filestore.aqa.org.uk/resources/mathematics/specifica...
Cf. 'S' (Statistics), 'M' (Mechanics, i.e. mathematics of physics), 'C' (Core), and 'FP' (Further Pure).
You're right that Computer Science/Computing is separate, but it's not a particularly common and certainly not a required choice for people pursuing CS, many view it as unworthwhile/a joke either in advance or retrospect.
Certainly when I went through they'd far rather you took as much mathematics as possible (which can be three full A levels - Mathematics/Further/Additional if you take all of the modules) followed by sciences. A faculty member at Cambridge expressed an explicit preference for Latin over Computing at A level.
This is how it's already done in the UK, and has been for decades. When I was in high-school (or rather the UK equivalent) about twenty years ago we did discrete maths, algorithms, and data structures in the regular maths classes. I don't think computers were even mentioned - it was all described as maths topics.
And this was a bog-standard state school.
So when I want to university interviews I was already able to describe for example how to implement a hash table.