First off, the UK government MUST go ahead with this even if they do not want to.
We live in a democracy in the UK and the people have spoken that they want to leave, the UK government has to follow that. As other people have pointed out, the referendum does not really count for anything and the government could just tear it up and not bother with it.
It's difficult to answer with 100% accurate because it's not something we have faced in before, with the government ignoring the people on this scale. But, I imagine that it would have serious consequences for them.
For example, it could cause a lot of people to protest, maybe start rebelling against the government and force them to dissolve parliament. As people would lose faith and confidence in their government that they elected to listen to the people. It would, end really badly for them.
As for the party in power (Conservatives) they would most likely not exist after this. I imagine that it will be very damaging for them. I'm not saying that it would destroy them completely however if you look at the previous general election, the Labour party took a major hit and did not do anything on this scale. People react differently and come down hard on the people in power. Let's take the Labour Party at present, a lot of blame is given to Jeremy Corbyn for his leadership skills following this result. I imagine that the same would happen if for example Boris (If he becomes PM) did not follow through with it. His party would simply rebel, resignations would follow, causing the party to be in a crisis which would eventually result in him losing his position. When things like this happen, it does not just effect the political party but if something like this happens within the party that is in power it could have a huge economic effect as well as a security threat. People see the UK is unstable then this might show weakness and with the conflicts and uncertainty within the world at the moment, this would be very bad.
In the EU it could also have serious consequences, for example the deal that the Cameron made with the EU has now been destroyed and is no longer on the table (This was part of the deal that the UK voted to stay in). This would mean that the government would have to go back to the EU and try to make a new deal, which, if you think about it, the EU would have the upper hand as it's kind of like saying "Ok, we have gone outside and it's too cold, we want to come back in" -- The deal would then be less on the UK's side.
It would also put pressure on the leaders of the EU from other members. In many cases, a lot of the other member states within the EU do not really like the way that the Brits get "special treatment" that some other countries do not get. Why should the UK dictate, throw themselves outside of the EU and when they do not like it, everyone bows down to them and let's them back in?
I believe that peoples minds are made up, whether they knew what they were actually voting for or not. The EU has (kind of) made their mind up and wants the UK to leave as soon as possible. All that's left is the UK government to "push the button" which is providing more difficult than it was first thought it in the campaign stages.
Either way, it has been agreed by the UK people that they want to leave, it has to be followed up by the government AND there will be no referendum within this government. That being said, legally, there's nothing to stop another government getting into power and holding another referendum (As I understand). There is a very high chance that there will be another general election in the near future.