NO Internships, Target

Here’s my situation. I am a student in the UK, I did my undergraduate in Chemistry at a non-target and decided it wasn’t for me. Now I am about to do a MSc in Economics (1-year) at LSE (did a diploma in economics this last year).

I am very interested in management consulting, but I have no experience at all! The only economics related internship I have done was a small stint at a think tank. How far can reputation of a target take you? I assume I should try find an internship position rather than a graduate programme for this summer, but it is difficult given many of them recruit only penultimate year students. Does anyone have any advice?

Thanks!