You'll want to drill the nylon with a bunch of appx 1/2 depth holes to give glue more bite. No glue will hold well to nylon when it flexes, but you can improve your odds.
For smaller pieces I would say E-6000 craft glue because it sticks decently to fiberglass and nylon but can be rubbed off of the surface later. unfortunately the large area would possibly take years for it to completely cure since the solvent has to escape.
Epoxy would hold with the drilling, but as others noted you will have an epoxy coated transom.
The 3M gray auto trim tape (or any of the other ways they market the same gray foam tape) sticks OK to nylon but not as well as to other plastics... The ABS typical of auto trim pieces it does work well if you are sure to dewax/degrease the plastic to get the mold release off. Similarly you'd want to ensure there's no mold release agent left on nylon (or the fiberglass) for best results.
Best is to drill the transom.
Drill as if you were going to mount it... (you can go ahead and put in screws)
Pull the screws and the item being mounted off.
Drill the transom holes at least 1/4 inch oversize diameter. Epoxy in a dowel that is a loose fit in the hole. Drill and put the screws into the dowel to mount your device. You'll have an epoxy barrier to keep the water from migrating into the transom from the holes.
