Well first of all, it depends whether you're wanting Classics, Classical Civilisation, or Classical Studies.
Classical Civilisation and Classical Studies don't necessarily require you to take any language modules during your degree, so in theory you can get away with never studying Latin or Ancient Greece at all by picking all non-language modules (although you can choose the language modules if you prefer - I'm doing Classical Studies and I'm opting to do 60 credits of Latin as part of my degree).
Classics almost certainly will require you to study some Latin during your degree, however as not all that many schools offer GCSE or A Level Latin, universities don't automatically assume that you'll have previously studied an ancient language.
Cambridge states that if you have studied A Level Latin, you can study your Classics degree with them over 3 years, whereas if you have not previously studied Latin, your Classics degree will be over 4 years.
So, in short, don't worry too much about not having studied an ancient language before.
As for not having taken a modern foreign language, that shouldn't matter at all.