As someone who's applied to LSE Economics and had an interview for Imperial's Economics, Finance and Data Science this year, I would say apply. Your A Level Predicteds are very good, and A*A*AA inc. Maths and FM definitely makes you a competitive applicant. As with GCSE's, as long as the 5's or 3 isn't in Maths or English Language, and instead in some irrelevant subject, you should be fine - Most other applicants might have better GCSE's, but with your contextual circumstances, that shouldn't be a big deal and your A Level Predicted grades with extenuating circumstances are more impressive.
So don't steer away from LSE or Imperial, because as long as your personal statement is strong and has relevant supercurriculars, you're in a good position as top unis (especially LSE) absolutely scrutinise the personal statement and it is the main determining factor ie. If you have a better PS than another candidate, even if he has better grades, you will most likely receive the offer. If you decide not to apply, then you might regret it in the future, as to why you didn't try.