To be clear, my opinions may be biased cause I graduated from UK. That said, you still have at least 5 years from now until you are qualified. Things can change a lot in 5 years.

I won't say it would be easier to get into an intl firm with a UK degree as compared to NUS degree - both are equally recognised. I have friends who graduated from both local and UK unis getting into foreign firms, although more of my UK friends landed jobs in intl firms.

If I were in your shoes, I would choose LSE if you are able to afford it. Living in another country broadens your horizons, and strengthens you as a person. We study less subjects in UK, typically 4 modules per year (whereas NUS teaches i think around 8 - 10 subjects a year? - I am not sure), and we probably need to read up on other aspects of law (e.g. insolvency law, intellectual property law etc) if we did not choose these subjects for the final year. But you also get to have lots of fun, travel extensively and you get to put yourself out of your comfort zone.

I can say the 3 years in London changed me as a person, in a positive way. I learnt to be less competitive and that you don't need to step over another person to climb your way up I don't know if it's because of my UK degree that landed me the job, or the person I am now due to those 3 years in UK.

If you are able to graduate with a first class from LSE, you may be able to land a training contract in one of the MC firms in London itself So that opens another alternative for you.

Just enjoy the journey. Many further hurdles to overcome (e.g. getting a training contract, passing the bar exams, being retained etc). Work will solve itself out eventually and getting into an intl firm isn't and shouldn't the end game (imho). Good luck!