Typically universities only NEED to see passes in English and Maths. More selective and/or popular universities usually have a nominal requirement to have taken and passed at least 5 GCSEs. Some courses have more specific GCSE requirements across the board, primarily medicine/dentistry/vet med. Some courses offered by specific universities have GCSE requirements due to the nature of the course and it's popularity, for example Maths at Cambridge.
Both Oxbridge universities will consider GCSE profile, however Oxford is the only one which tends to actively make use of it through the majority of the process, as it's used in part to determine who they shortlist for interview, and then they consider it post interview in making offers. Cambridge on the other hand interviews most applicants and then heavily considers interview performance to shortlist candidates post-interview, where GCSEs are considered to an extent.