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Congrats on your admits!
Unfortunately neither school is really going to put you into London. Though if I had to choose, I would probably pick Georgetown as it is better known. Though again, it is not McDonough but Georgetown that are known and you are likely to find the alums in London. You may have a shot and a chance though every english-speaking alum of INSEAD, LBS, IESE, Judge, and Said are going to be angling for London consulting opportunities. There has to be a reason for you to stand out among the sea of applicants... applying from McDonough may be just obscure enough that you actually can stand out but it is also a moonshot to an extent.
Owen is great for healthcare and is much stronger for healthcare play. I am not sure how much you will love living in Nashville vs. DC. DC is definitely more cosmopolitan and the center of everything there. Owen is also very strong but in Nashville. It's been growing and lots of people been moving there but you still have a fair share of Southern culture which some may find particular. I would not worry about it too much though

I would really be curious what
rtbs15 Dleerium think about applying to London for consulting from the US. How ambitious is it?
I don't think that moving to London from McDonough is a realistic goal. London is one of the toughest 3/5 offices (depends on who you ask) to get into for any of the major firms, and they have their spots well covered by core schools. For the US, those core schools would be M7s, although there isn't much appetite to move to the UK with US debt.
Still, I'd say apply to all the firms you're interested in and put London as your top choice (+ start networking YESTERDAY), construct the story as good as you can, and you'll find soon enough if it worked or not. If it doesn't, you'll get to your second/third options, but make sure that those options are offices with less competition, as you'll find the same issue with NY for example. McDonough is not a core school for top MCs.
Just to clarify, I'm not saying it's impossible, but your chances are low. If it's your dream go for it, but I would hedge my bets with my fallback options.
Good luck!