The petition is at
2.7 million signatures in a bit less than three days.
I suppose they are unlikely to get 16 million, but they are certainly making progress. At this point, a good many parliamentarians are probably looking at no downside to debating this.
As you said, it was close. If the events of the last two days have been enough to change the minds of 2% of the electorate, then any hypothetical second referendum held tomorrow would be in favor or "remain".
So here is the question: if the referendum was very close (as it was), and polling data starts to show that the majority of the population no longer supports it, what should the government do? After all, the referendum was hailed as being non-binding. Should the government continue to implement policy based on a non-binding referendum if they have data indicating that public opinion has already swung away from the results of the referendum?
This is not a far fetched scenario. This was a very close vote. Anyone assuming that this is inevitable seems to act as if there were some sort of super-majority victory. There was not, a second poll held today could easily have different results.
ETA: If anything, the public seems to have been misled as to whether or not this actually was "non-binding". In practice, all parties involved seem to be treating the results as set in stone. It seems to now treated as entirely binding.