To a great degree, your trip weather will depend on your route. For example, it can snow any month in Yellowstone, and it has snowed on us there in June. The chances of snow in that area, and I imagine other elevated areas, goes up dramatically in mid to late September.
In the Sierra, likewise. Two weeks ago while in Sequoia NP, the main road through the park was closed to vehicles without chains (us), due to an overnight multi-inch snowfall that was not predicted. One of the major roads in Yosemite was still closed at that time because of snow, another closed due to a major rock slide.
We have also been partially closed out of other parks like Crater Lake and Glacier in the "summer" due to snow.
So in general, I would agree with others who have stated to go north at first, then work your way south to complete your loop. Be prepared to deal with weather issues, e.g. take warm clothes, as it is part of the adventure. Have a great trip!