Would I be able to get I to St Andrews or LSE with these grades?

As long as it meets their entry requirements and your personal statement is topnotch then you're fine. Not many look at GCSEs as a large determining factor for giving you an offer. Your predicted grades are there to see if you can meet their entry requirement and they will judge how likely you're gonna get those grades based on your references from your teachers and your personal statement. From my understanding, LSE places really high stress on your personal statement being one of their main determining factors as they don't do interviews, entrance exams etc, same goes with other unis that only use your UCAS application.
Rememeber offers are not determined solely by grades, your personal statement and references carry way more importance (in the assumption you've applied to courses where your predicted grades meet the entry grades, if they don't meet the grades listed on their website, they likely filter these applications out into a rejected pile since you didn't research well enough)

Thank you so much, I’ve looked at the grade requirements and I meet all of them, what would you recommend to do to strengthen my personal statement?