In regards to people saying you have to study engineering, maths degrees can give you a great baseline for many jobs that are stem related. What you will study might be highly theoretical, but maths graduates are sought after for their problem solving abilities and are known to generally be pretty smart. Im a second year maths student at Bath for example, and Bath is known for its year-long work placements that students across a variety of disciplines get. Combine that with a maths degree and you should have really good job prospects, and experience in a field that you're interested in so you can get a feel for it. So remember that it is worth looking for internships/placements while at uni.
Not sure about Oxford but I will say as far as I am aware Cambridge don't care that much about personal statement, though it still needs to be decent. Offers are much more based on how well you perform on the interview at Cambridge, and the part that's really hard is actually meeting the offer of grade 1 in STEP 2 and 3. If you're serious about Cambridge, I would start to look into preparing for STEP as early as possible, given you've covered the content in a given question (STEP database will help you with that).