I might offer a different perspective and reasons. I have also considered staying wheee I lived before and commuting to London for university and I am so glad that I decided to move to London. Granted, I am a postgrad and my course might be a bit more intense than for example first year undergrad. BUT! You still will need a lot of time to study and it is very full on and tiring. Add a long commute on top of that and you might be extremely tired after just the first couple of months.
It sounds like your reasons for moving out are based on getting a “university experience” and independence, and honestly this is also very important, but another factor to consider is how close to the campus you are. If you find it difficult to study and focus in your family home, you might struggle with commute times if you are planning on doing your work on campus (train times, travelling late on your own). Also, if you would like to get a part time job while you’re studying, there are plenty of options in London and some very flexible jobs that will work with your schedule. The time you spend on commute could be spent on getting a valuable work experience. Especially if you’ve never worked before, it’s always nice to have some experience with transferable skills on your cv, be it an events or a retail role, it all matters!
On the other hand, staying in your family home is a great way to save money… it is SO expensive to live in London, no matter how much student load you get.