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Don;t know about Melbourne. LSE is choosy and needs you to have good academic credentials. UCL above average, and SOAS is more like if you can pay and write a decent SOP (or pay someone to write one), you'll get in.
Lse, ucl and soas are selective but not too difficult to get into, if you have your finances sorted. Scholarships are difficult. No clue on melbourne.
Idk about Melbourne but the others - I’m willing to bet for public law type fields if you’re somewhat proficient in the jargon and the oft repeated virtues then you should be able to get in no problems. Requirements aren’t very tough. What is difficult is getting funded.
lse - ucl - need to have 65%+ marks during application

soas - anything more than 60%

if you have that marks and can pay, they will take you