First of all: Welcome to the board, FederationCitizen
And congrats on your thread starter. A very interesting analogy, indeed.
Legally, the vote is not binding. Theoretically, the British government can ignore it. On the other hand it'd be a sort of own-goal to first ask the people and then ignore their vote. Either way the current government is not likely to get re-elected.
I was a little surprised by the swift reaction of the other European states. It's kinda "you want out? ok, get out right now". Usually, decisions take a far longer time to be taken.
However it gets understandable if you look at the prob from the European point of view: ever since they joined, the British demanded - and got - special rights. Same currency? No thanks! Trade restrictions and taxes? Surely not for us!
For years it's been chafing the other countries in the community that Britain got all the advantages without having to endure any of the drawbacks the others had. Even if the British government chose to ignore or to repeat the vote - if they want to stay in, they'll inevitably have to make sacrifices where these special conditions and privileges are concerned.
And that, I think, is what's behind the suspiciousely swift "ok, get out" of the other countries. GB is in check and has to sacrifice not only a few pawns but a bishop or two to get back into a safe position.