From a risk perspective I would highly advise against this. Each region has a stake in their interns and invest a lot of time/money in filtering out/recruiting you. I would highly advise that you look to convert that internship into a full-time role in your home country. It's hard enough to get into BCG...let alone when playing games!
Now, rest assured, you can absolutely switch to London after a few years working FT in your home country.
I've transfered a few times within companies and have the following advice:
1) Build a stellar reputation - do good work, work hard, and be known as "the guy/gal" for xx
2) Build your network - network agressively (yes, networking doesn't end once you've gotten the job offer). Make sure you're known by and have allies in people who make decisions such as staffing managers, Partners (specifically those responsible for recruiting/resources and who are heads of industry/function verticals), etc.
2) b). When I say network "aggressively" please don't be needy/annoying :) There's nothing worse than someone who is obviously working the room or trying to please!
3) Look out for opportunities - as mentioned in this Q&A already, look for office transfers, short-term projects, ambassadorships, etc.)
4) Practice patience and be flexible - how long have you tried for? This might take a year. And you need to be ready to go at the flip of a coin.
Bide your time, keep pushing, be smart, and you'll get there!