In regards to foreign policy, Europe will really have to invest in its military capabilities if they want more independence from the US. The fact that Europe was not able to effectively intervene in either Bosnia or Libya without US assistance when they are literally in the EU’s back yard means that the EU foreign policy will play second fiddle to US foreign policy.
There are of course reasons for this, and obviously Great Britain is acting in their self-interest (they don't want a "shadow NATO" and they don't want to diminish in importance in USAs eyes, UKUSA/FVEY etc.), but Great Britain's interests are not really in the interests of Europe as a whole.
Now that Brexit seems to be happening, one way or another, expect to see a lot of European Union movement in the defence area. Also, time will tell if the e.g. UK Joint Expeditionary Force will stay in existence and enhance the new pan-European defence mechanisms with, among other things, British nuclear weapons, or not.
Furthermore, it is very naïve to assume that Europe is not already doing full-on preparations for the military implications of Brexit as well as the everso more likely dismantling of NATO. Mr. Trump has been signalling the latter for a long time, and even though much of his output seems to be a clumsy way to lure EU into doing more military business with the USA, the fact is that the White House and particularily the President of the USA is acting more and more erratic every day, so basically anything could happen (and probably will!).
Having a strategical autonomy and owning a substantial defence ability and capability is an existential issue for Europe, given its geographical location, as well as the increasingly hostile political rhetoric and gray zone actions of its neighbours, as well as the continuously erratic behaviour of its number one ally.
North Korean can't be invaded, because they have nukes. Iran would be untouchable if they had nukes. The Soviet Union had to obey the orders of the US, until they got nukes of their own.
Sweden, of all countries, had a greater effect in the Albanian genocide than the US. Had Srebrenica been controlled by the Swedish armed forces and not the Dutch there would have been no massacre, or at a minimum the massacre would have been opposed by the armed forces stationed there, which would mean that it would have been no massacre at all.