When the word establishment is used in a political sense, it "generally denotes a dominant group or elite that holds power or authority in a nation or organisation." (https://en.wikipedia.org/wiki/The_Establishment)
So in this case, the phrase refers to the highest levels of government or political leadership, both within Britain and within the EU. It also acknowledges that the elites from Britain and from each of the other EU members are a part of the single entity that is thought to hold a grip on that power.
Finally, I don't think that the article is assigning responsibility for Brexit to any party. Rather, it is pointing out that the Establishment failed to anticipate the passing of the Brexit referendum. In the same manner, the Establishment may fail to anticipate problems during the actual Brexit that will cause significant issues to investors. The article that you linked to cites text from a larger discussion of Brexit's potential impacts. The text immediately following what you've quoted reads,
A chance remains, however, that negotiations break down completely,
Reid's team said.
"The real financial crisis could arise if the UK experiences a
dramatic ‘hard’ Brexit with relations completely breaking down with
the EU. This would not only have economic implications for the UK and
the EU but also on geopolitics."