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.