Harry and Ron each earned seven O.W.L.s. Hermione earned eleven – ten Outstandings and one Exceeds Expectations (HBP5).
The only grades below “Exceeds Expectations” that Harry got were in History of Magic (he passed out during the exam), Astronomy (the exam during which Hagrid was arrested in full view of the examinees), and Divination (lost cause). Harry realizes that Snape won’t allow him to take N.E.W.T. Potions with only an Exceeds Expectations grade and, therefore, he won’t be able to become an Auror (HBP5).
Ron says "it's been a month" since O.W.L.s finished, and the exams probably finished on 18 June (OP31). We also know from HBP06 that Harry spends "a few weeks" at the Burrow prior to his birthday on 31 July - so it's likely that it's been a little less than a month, rather than a little more. That, plus our date reference from HBP18, means that this is most likely Saturday, 13 July.