I live in northern Wyoming and have cycled nearly every paved road in the state.
(And quite a few dirt roads, too.)
My guess is that it will be about mid-June when you hit Yellowstone - pretty perfect.
Yellowstone can still have a late snow then, but you should also catch peak wildflower season.
If you are willing to do some serious climbing - then crossing the Bighorn Mountains will be incredible.
I have mapped out sections across the northern part of the state -
From Yellowstone Lake - east on US 14/16 to Cody - the Wapiti Valley is a stunning ride.
From Cody stay on US 14/16 - but cut over via Wyo 30 to Basin - high desert and irrigated farms.
From Basin take Wyo 31 to Nowood Rd. to Tensleep - incredible colors - like a Gauguin painting.
Up from Tensleep on US 16 - magnificent canyon walls and over to Buffalo.
You should really plan to camp at moderately high elevation and take in the wildflowers here.
From Buffalo, US 16 to Ucross and continue on US 14/16 to Gillette.
(A lovely ride - longer, but soooo much better than getting on I-90.)
From Gillette, Wyo 51 to Moorcroft - then US 14 and Wyo 24 to Devils Tower.
Continue on Wyo 24/SD 34 thru Aladin (general store) to Belle Fourche.
Tensleep Canyon
If you are interested I can forward you the strip maps.