Sponsorshop options?

Hello all,

I am in the last year of my four-year undergraduate degree from the top university in my country (which is either U.S. or Canada, don't want to give it away). I applied and was admitted to Oxford's BA in Jurisprudence as a Senior Status student. I know how law works in the U.K. and the stuff with the vacation schemes and training contract and that I'd need to hit the ground running in applying for those when I get there.

My question is, considering I'd have two degrees under my belt and one from Oxford, how likely is it I can get sponsorship at a good firm in London? My goal for a while has been to relocate and settle permanently in London, but it would be a huge risk if sponsorship is scarce and I return to my home country with a degree that doesn't qualify me to practice law. Just wanted to get people's thoughts and experiences with this to better inform my decision. Thank you all!

Edit: Sponsorship* not shop.