What are my chances of getting in to Oxford/LSE/Imperial MSc?

I'm doing a quantitative degree at a UK university that's ranked at ~40 (although it's in my hometown in scotland, so I literally pay nothing.) I'm fairly sure I can get a first. My school grades were easily more than good enough for Oxbridge etc. My work experience is nothing but a few voluntary roles and a 2 month non-finance related internship.

Do I have zero chance or is it worth a punt in applying this autumn? Is postgrad a harsher playing field in terms of applications? Oxford basically give any half decent school pupil an interview for undergrad; is it the same here?

Of course it's worth a punt whether my chances are 0.01% or 10 %, it's just I'd get embarrassed about asking people at my current university for references and then failing.