Hi jacket882,
1) It's difficult for me to say, as I don't know how cometitive recruitment in London is. I can say that the offer-to-first-round-interview is around 10% (maybe something less) at MBB offices in Italy (first-interview-to-CV is 10% as well). McKinsey and Bain's offices in Milan are huge (350 people), possibly bigger of the London ones, while BCG has fewer people (about 100), so the opportunity is there.
Despite what every HR will tell you, the bar varies a lot across market cycles: being hired now is almost impossible if you're not part of the MBA round, and even so quite competitive.
If you are an international and you can speak Italian fluently, you stand a huge advantage. If you are not English, that's even better since you would know another European language fluently (ideal would be Eastern Europe). The thing is, you have to be fluent in spoken and written Italian, even if you are not impeccable: if not, you hardly have a chance to be extended a full-time offer (Italian is the language of most engagements).
2) GMAT - If you have an above 700 score, report it on your résumé. If not, don't. No one will ever ask you for that nor having a 790 will matter past the CV selection step. It all boils down to how well you do in the case, 750+ scores who can't asnwer a case in a simple and straightforward way are rejected on first round on a regular basis.