I have experience of this.
For a small number of barristers in the highest sets, personal wealth will make no difference past the BVC as the chambers offer a good salary for pupillage and all but guarantee a good work stream. For those chambers, it is extremely meritocratic. To get there, you need more than a first - you need the highest intellect, judgment and common sense; excellent confident advocacy skills; and all of this provable through scholarships, prizes, or world experience. It is unthinkably competitive.
For other smaller chambers or regional chambers it’s hit and miss due to the self-employed ‘eat what you kill’ nature of the role. Hard on women to take mat leave. You can have a great life and not work considerably more than in magic circle. Or you could have it very tough.
I’d only advise on the bar as a career if you are extremely ambitious, hard working and the sharpest tool in the box. As for criminal bar - like medicine or midwifery it’s a vocation and you certainly don’t do it for the money.
As to where you got your degree from: obviously there’s a statistical bias towards oxbridge due to the sheer number and quality of candidates. It’s harder to get into, it’s tougher to stick it out. It does say something about the candidate. Nonetheless, a shiny first from Bristol with scholarships, moot prizes etc behind her/him, excellent interview etc can well make it, as can people with other relevant life experience.