Breaking into IB in London

Hi, I am trying to break into IB in London, UK and would appreciate your thoughts and advice on the best way forward. 

I am currently a Senior Associate in the Corporate Finance division of an investment bank in Malaysia. i think its equivalent to a 1st year or 2nd year analyst in London or wall street. For context, a typical corporate finance dept here provides capital markets and corporate advisory services which includes IPOs, M&As, restructuring and so forth. 

I have been working as a senior associate for about a year and have been involved in one IPO and am currently involved in a couple of M&A deals. Previously, I was a Management trainee in at the same bank where I rotated through Client Coverage, Equity Research and Corporate Finance within the investment banking division. 

Academically, I did my undergrad at some small, unknown uni in Kentucky, USA and did a double major in accounting and business administration (concentrating in finance). I also did do an internship during my junior year which was a 3 month program at asset mgmt firm in Malaysia. 

I'm now looking to move to a bigger market to gain some exposure to larger deals or projects. I understand that my uni background is weak given that it is a non-target school and alumni connections in IB or big banks are very limited and unresponsive lol. I did come across a few posts on reddit and even WSO that said that doing a Masters at the right schools would improve your chances significantly. 

Given all this, I am considering whether doing a Master's at a good school in London or UK is the right move? If so, which schools and master's programs should I look at? and is this path even worth considering or is there a better path to break into London's IB industry? 

Would really appreciate to hear your thoughts and advice on this.