Usually the largest flowers tend to take the longest to bloom, but its not an absolute rule. Some plants simply come one much faster than others. I have a couple of medium sized dahlias that usually flower before some of the poms, and one larger flower that comes on before most of the mediums, but as a general case that is true.

Planting at the end of May should definitely get you some blooms by late July, early August. In my dahlia patch, almost everything is in full bloom by mid August, and I am cooler and probably shadier than you are. (I mean my yard - not me)