LSE, Cambridge and Oxford do not generally give scholarships to international students, although Oxford and Cambridge have some specific ones for specific countries- you can check and see if yours is one of them.
There are 2 types of scholarship: merit aid and financial aid. All of the US colleges have a lot of info on their websites about what aid they offer and who is eligible for it. Note that Harvard, Princeton, UPenn and MIT only offer financial aid, not merit aid. Duke has a small number of ‘named’ merit scholarships, as well as financial aid and CMU has some merit scholarships as well.
Be aware that getting some amount of scholarship may be very easy- but the amount may not be very big.