Does the fact that I don't have Oxford or Cambridge universities on my CV count against me when I am applying for pupillages?

Not at all. While individual chambers may have slightly different selection criteria depending, for example, on the type of work and practice that they undertake (for example whether they are looking for jury advocates or purely academic lawyers), the university at which you studied is not going to be one of them. The Bar has a strictly observed equality and diversity code which should ensure that any unfair bias in recruitment, including towards certain universities, is eradicated. In complying with this code, chambers should use objective criteria in assessing candidates both at the application stage and in the interview stage, both of which are scored and then separately monitored to ensure strict compliance. The more widely used selection criteria might include, for example, showing commitment perhaps through work experience or having undertaken mini-pupillages, demonstrating both intelligence and a real desire to enter the profession, as well as a strong academic record.