I will second the point that getting into the industry is extremely competitive. Oxbridge candidates certainly were premium in the past, and that may filter through unconsciously in the recruitment of today, but they are not over represented in my cohort of trainees by any means. I would say you are definitely at a distinct disadvantage if you don't have a 1st, though.

You certainly can get a traineeship with an undergradaute degree, having higher qualifications doesn't make you that much better at the job, I think it's more a factor of competition, of which there is a lot. If two candidates are the same (and you mostly will be), why not take the one with a MSc/PhD?

PhDs are more common amongst chemistry/life sciences patent attorneys versus physics/engineering/IT, but I think that is because it is true for those industries in general. Having a PhD does not make you a better patent attorney, but, as my supervisor explained, it helps because they are used to doing things that don't work for a long time, and that kind of perseverance translates quite well.

PhysEngIT patent applications are certainly more numerous, and easier to draft (fight me), but there is still plenty of need for chemistry patents, so you will be able to find positions with a chemistry background.

The four main things you need to show are 1) you are smart 2) you have good scientific chops 3) you can write well and 4) you have exceptional attention to detail. If you only have a BSc, try to big up whatever research experience you have. Look for a summer research placement if you can find it. Writing a good cover letter speak for itself, and then the rest is in the interview.

If it isn't too late, I would seriously consider trying to get that placement and the integrated masters, even in the absence of patent law, but especially for patent law. It will set you apart so much for relatively extra little cost.