For anyone interested in using their brain right now:
July 16th
ETA: Albert was not told the month of May or June, because each has a date (18 & 19 respectively) that would tell Bernard exactly when Cheryl's birthday is.
Now Bernard knows that the birthday is in July or August and mentions that he knows the birthday.
Bernard was obviously not told 14 because he wouldn't know the correct month. So we're left with July 16, August 15 and August 17th.
Then when Albert responds that he knows when the correct birthday is, it means that the month had to be July, because Albert can only be sure of the correct date if it was July. Because in August there were 2 possible options and July only had one. Alberts confirmation that he knew the correct date points to July 16th as the correct date.