ahumne:
That consultant is probably right. He or she is not saying you can't get in but that's it's a stretch. Oxford and Cambridge are amongst the top MBA programs in Europe. For most people, it's considered a reach. If you only apply to those schools you are running a risk that you will be swept out of choices.
You have some good experience overall but everything feels somewhat thin. Your work experience is a bunch of short experiences. Six months in a gig is not going to win you too many points. You are competing with candidates who have been in jobs for multiple years and risen in the ranks at organizations. That's not the only criteria but it's one that you have to take into account when sizing up the competition. The same can be said of your extracurricular activities. They really aren't too robust.
Lastly, tech consulting isn't exactly an inspiring career choice. That's not to say you won't come up with something interesting to propose to the AdCom at these schools, but it's not too imaginative and will potentially downplay interest about your candidacy.
Again, I'm not dissuading you from applying to Oxford and Cambridge but it's probably a good strategy to also apply to schools like IE, ESADE, St. Gallen, etc. that are lower ranked and will still open up professional doors for you.
Hope that helps!