If you are applying for a three-year course at Cambridge you will have to do a 60-minute translation exercise, whereas if you are applying for a four-year course you will have to do a 20-minute language aptitude interview.
As for Oxford, four of the six courses require the Classics Admissions Test (there is no Admissions Test for Classical Archaeology and Ancient History, and Ancient and Modern History requires the HAT). Classics and English, Classics and Modern Languages, and Classics and Oriental Studies all require a second Admissions Test as well.