I imagine this would be similar to Texas deciding to secede from the US. Something largely unthinkable and ultimately somewhat disasterous for both parties.
It seems from the outside (can't get much more outside) that the leave voters were inspired by immigration issues and limited thought to the real and actual problems that would be involved in leaving the EU. The initial drop to the value of the pound is one issue but for effective trade to continue in the EU it seems like all the things the leavers don't want (free movement of labour etc) will have to be agreed to - which makes it seem like a moot point.
Obviously the issue must be much more complicated and the British public must have a better idea about this (the world hopes). From the outside it looks so heinously misguided and self-desctructive it's troubling.
Using Texas as an analogy is not so good; the US and the European Union are two separate political entities. The US is a federal republic and the EU is nothing more than a collection of treaties governing trade and border controls and currency with a clear opt-out for any state that wants to use it (Article 50).
As far at the financial consequences, they have been largely overblown in that the pound and the British stock market have already rebounded to pre-Brexit levels.