Hi,
I just finished my GCSE's as well (8 9's and 5 8's). Although I'm not the target audience you are probably asking the question too, I went on a trip to Oxford in January and have done my own research as well. These grades are good, and honestly I don't think you should worry too much. Why? Firstly, when asking Oxford, they said that while GCSE results are important A-level results carry much more weight. There have been cases where students who got just 6's and 7's have gotten into places like these, because of their progression journey - they went from OK grades at GCSE's to really good grades at A-Level. Another thing I would say is the importance of work experience. During my Oxford trip, chats to teachers, and information online, they emphasised that you need to show passion for your subject by doing work experience, reading books, attending events. For example, I want to do Economics, so I was on the Economic council for Model United Nations, read a book about economics, and I have recently secured work experience at Grant Thornton. However, with your results, you will have to work harder, and this is a fact. But as long as you dedicate yourself to your work and don't burn yourself out, it will be OK.
Although these universities are good, if you don't get in it isnt the end. From my tour guide (who was a student at Oxford) on my Oxford trip, he told me that there are a lot of good Unis out there that aren't as well recognised. This is because Oxford and Cambridge are a brand, and everyone recognises their name immediately. My friend's brother got accepted into Cambridge, but he actually went to Bath instead I believe, because he felt the course was better. For me, I don't want to go to Oxford because they only do Economics AND management, which is not really what I want to do.
So, please don't stress too much. Your grades are good, you've got 2 years to do well in A-levels, and the most important thing is getting on a course you want to do.