Walking a wet asphalt shingle roof isn't any more dangerous than walking a dry roof, provided it's clean. Yep, you read that right. I said it.
Of course, there's a bit of risk involved with walking any roof, but the added risk caused by walking a wet asphalt shingle roof is way overblown. I've been walking these roofs wet or dry for as long as I've been doing home inspections, and I've never found wet asphalt shingle roofs to be any less safe than a dry roof.
Again, I'm talking about clean roofs. All bets are off when the roof is dirty. If the roof is covered with things like moss, algae, leaves, or pine needles, it's definitely less safe to walk when dry, but especially when wet.
But that's just anecdotal. Me not falling off a wet roof does not a point make.
Grip Testing
To demonstrate this, I did some testing with wet and dry shingles. I took a new 3-tab asphalt shingle and tested the gription (not a real word) of several different shoes on both a wet surface and a dry surface. I used a digital crane scale to pull on the shoes with 40 pounds of weight applied downward.

I wanted to use more weight, but it was too tough for me to gently pull on the scale with more weight applied. I tested several different shoes; Cougar Paws (of course), Slip-Resistant Sketchers Work shoes, casual Sketchers shoes, and some old Nike, Adidas, and Fila shoes.
I first tested with dry shoes and recorded the maximum amount of weight that I could get with each set of shoes. I then soaked the shingle with a wet towel for 10 minutes and repeated the test. In each case, the wet and dry readings were nearly identical.
There are surely a thousand other types of shoes that I could have tested, but the results would surely remain the same.