European here. Contrary to your belief (as probably induced by the somewhat biased media reports), many European civilians are not mourning at all. Many of us are happy to finally see EU dictatorship come to a halt. The Brits will do fine outside the EU. And so will many other countries.
EU is a failed project, perhaps good for the elite and large companies, but it sucks for ordinary citizens. Unlimited import of society-wrecking hordes, unreal money pits, total neglect for ordinary citizens, destruction of carefully crafted wellfare systems, pensions down the drain, job losses, and mind-boggling burocracy. That's EU for us. This project should return to being an economic powerhouse, without the common currency and without tens of thousands of useless, overpaid burocrats.
The Brits have taken the lead. Will it hurt? Sure! But better to cut the ties now then to stay aboard a ship heading for the cliffs. I expect other countries to follow. The Clash had this in their lyrics: "If I go there will be trouble.. If I stay it will be double!!"