Need some advice, MSc Econ Student

Hey, I did my undergrad in Chemistry in a decent but not top tier uni in UK, did pretty well + did a diploma in Econ and now I am about to do MSc Economics @ LSE.

I am interested in economic consultancy but I have no relevant internship experience, the only thing I have done is a little public policy think tank role for a few weeks. How far do you think brand name of LSE alone can take me? I guess the 'top tier' e.g. NERA, Charles River is definitely out of my reach, given I am competing against people with stronger econ background + internships?

Would appreciate any advice! Seems there are a lot of options but I have little understanding of my chances...