Are my GCSEs enough?

We do have AS Levels, they don't count towards the A Level anymore. You have to do the AS again with A2 at the end of Year 13. Also some AL's are being reformed some aren't. So they can make offers on AS levels because most schools will make you do it. I know people who got into good unis with worse results than that so you'll be fine.

My school are only doing exams at the end of year 13, so we won't have any A level grades when universities make make offers. The whole thing still sort of confuses me honestly

Which A Levels are you doing. One or more of them might not be reformed until 2016.
And I still think you'll be fine

maths, economics, religious studies and art

Maths and RS aren't being reformed this year so you will have AS grades in them, Economics and art are, so you won't have AS grades in them.
You can find this info on various different websites. Just type in A Level 2015 reforms in google or something.
Maybe they will take into account your Maths and RS AS grades to predict how you might do in the other two, so the unis will have some basis to make offers.