Hi, the applications are done through UCAS. Normally you'd sign up with your school and would upload your personal statement and predicted grades there. The school is also responsible for uploading some information too. You can apply to five universities and UCAS sends your applications to them. There's deadlines for when the universities want the application by and sometimes the school has their own deadlines too. I'm not sure how it is for overseas universities but I think the process should be the same.
Also have you begun preparing for the LNAT? I'm not sure about Southampton but UCL and Cambridge definitely require it.