Hi Op, a "global firm" is going to want someone who went to Oxbridge, or Durham, or failing that Bristol, Nottingham, Exeter etc. A global firm does corporate work, and your degree, in the current climate, is just not going to cut it.
I work as a Defendant, in house, and can see that the claimant firms, are increasingly desperate for work in the current climate. So you are going to struggle at the moment.
It's the top tier firms corporate firms who have the money to sponsor things like the SQE, so it's a bit of a catch 22 at the moment for juniors. And they will only pay if they really want the candidate. You are potentially throwing a lot of money at this, so think really hard at what you want from being a solicitor. Corpotate / global firms pay the most, have the deepest pockets, expect ridiculous hours and demand the best from applicants. High st practices have pretty much disappeared, and there are regional firms doing mainly private client work and some commercial, like Shoosmiths, Irwin Mitchell, etc.
Life in all theses places is very different, and if you get to interview you need to be able to show that you really want to work there, so you need to know in depth what the firm do and that you are able and willing to work hard and make a lot of money for them. FWIW my son has just got into Durham, and I have told him, not to go anywhere near law, as the rewards are not worth the effort, not for several years anyway, and not these days.