I'm not sure how the felt area of that table is attached to the table, but assume that it is attached by hardware under the table, or just sits on the opening. Once the top and old felt is removed, then add abd glue 1/4" foam to the wood then attach the speed cloth on top of the foam, stretch as tight as possible and staple under the wood. You will need to cut the overhang portion of the speed cloth in small v strips so they can be pulled tightly and stappled under the top so as to lessen or eliminate any wrinkles.
Once the cloth is stappled as best as possible, it may be necessary to remove and restapple in areas that may be wrinkled.
Speed cloth is fairly stiff, so a bit harder to prevent wrinkles on the edges of a round corner. Casino cloth is much easier to work and stretch out.
If you can't find the foam, you can glue together 4-5 layers of laminate flooring underlayment foam using 3M 77 adhesive. That is what I used under my cloth, and it is a really good compromise.
Good Luck.
