Breaking into management consulting in London

Hello everyone,

As the title indicates, I am trying to break into management consulting in London. For context, I study at a well known European business school, and have one M&A and one management consulting internship (both in continental Europe). 

Originally, I thought that the best way would be to study a Masters degree at a London School (LBS MFA or LSE MSc Finance to differentiate myself from large MiM cohorts), however I now understand that consultancies have a preference for undergrad students, and that recruiting for students in London graduate programs has been quite tough as of late. 

I am wondering if this is still a good plan to break in or if there is another/better way. I was also thinking about the HEC MIF/MiM and the Insead MiM, although they seem less likely to help me reach my goal.

Would appreciate any help!