Oxford/Cambridge do indeed have a smaller consulting funnel than, say, LBS/Insead (they're not target schools in the same way).
That being said, regardless of which MBA you're in, the steps to optimize your chances of securing an interview are the same!
How do you improve your screening chances?
1) Gain experience (internships, bootcamps, classes, stratups, extracirriculars, etc).
2) Network - speak to multiple people across offices and aim to get a referral
3) Review your resume - make sure your formatting is sharp/consistent, key words/experiences pop, and your experience bullets are properly written.
In regards to networking tips:
a) Reach out to people in your network
b) Reach out to people once removed from your network
c) Reach out to people with a similar backgorund to you (i.e. same alma mater, same historically underrepresented demographic i.e. gender, orientation, ethnicity, etc., same career switch, etc.)
d) Tailor a message to them specifically both showing interest in them and their journey and demonstrating that you have done your research and could be a valuable hire
e) Play "tag" across calls you get so that you can work your way towards the company/office/role you want
f) Never directly ask for a referral, but "hint" at needing one (this is nuanced and important...happy to talk through wording)