The "globally influential" and London being a top financial center will certainly decrease after Brexit. You'll see a lot of EU banking moving from London to Berlin or possibly Paris.
(Yes SV became less popular in recent year, but because of other reasons).
Its about the credibility of the financial system.
Some of the senior people at BAFIN should have resigned by now
Frankfurt has the draw as already established major centre, but Amsterdam, Dublin and Paris all got things going for them.
Recreating a global financial centre like the the City isnt a matter of just moving to another location in Europe. If it were that easy the Eurozone would have done it years ago.
London has the infrastructure, contacts, mentality, depth of knowledge and experience that the other capitals in the EU cant match. And why would they want to, the city is happy to do business with them. Why would the EU want to wreck a European financial centre that provides them with so many benefits?
They will be outside the portability regime though, so some trading will of necessity be moving.