First off I will say this about rifle slings. About every cheap alternative to appropriate type of leather that could possibly work has been tried and found wanting. Even if they approach the functionality they look cheap. I can't for the life of me understand why someone will pay good money for a rifle or whatever then put something other than a quality leather sling on it.
This said a good leather sling shouldn't cost more than $60! The
Montana Gun Sling is one of best I've ever had the pleasure of using and it costs under $40. So for the cost of two pieces of chinese nylon you can have one sling you can be proud of! NO BRAINER!
I can say this about fabrics. Any kind of knit fabric wont work because it will stretch too much. Woven fabrics of strong material like nylon (like a seatbelt basically) will work alright. They won't have any nice grip like the flesh side of a leather sling will, though. Also, all fabric is made of thread so all the edges must be finished very carefully otherwise it will begin to fray. Leather, as a material, doesn't have this very significant problem. Even ripstop fabircs and seatbelts can fray with just small cuts.
Unfelted wool as a material changes tensile and dimensional properties greatly with water, temperature, and heat. Felted wool, which is what cowboy hats are made out of and would possibly work as a sling, costs a LOT more than even the finest German Latigo leather which is best for making slings.
Any fabric made from a cellulose fiber (linen, cotton, hemp) will be terrible if it gets wet. Proteinacious (wool, silk) or synthetic fibers are better in the face of water.
One thing that could work is a leather sling with a felted wool pad on the shoulder under a leather sling. That would be nice.