It sounds fairly unlikely.

RJ answered a few questions about Nym. They weren't born, for one, they are genetic constructs, hybrids from various genetic sources, like for instance a gholam. Your theory rather have the Nym able to reproduce through a fairly twisted(and not all that logical biologically) process: the Nym bears a fruit (an acorn), which becomes an oak, which in turn one day produces tons of acorns that become Nyms. IMO, it's far more likely the Nym were partially made of oak-DNA, and the fruits of a Nym produce.. an oak that probably isn't much different from a normal oak (hybrids don't breed true). Chora trees are another hybrid construct, and we know they are sterile and to get new chora trees you need to make more, or reproduce them by cuttings (which are clones of the source Chora tree, genetically).

It sounds very likely the whole idea of trying to get constructs which can breed is a notion introduced by Aginor for Shai'tan. Constructs which can breed or seed would end up upsetting the ecological balance, and the Aes Sedai would be aware, and cautious about this. Of course, Aginor didn't give a damn about this.

RJ also answered a question about their souls. He said they were constructs, not natural, so they had to "borrow" one from the soul pool (it's part of the process of making them, I guess), but unlike souls "borrowed" for Shadowspawn, who are so tainted they couldn't be reborn again as anything but shadowspawn (so these souls will be in effect "damned" never to be reborn if Shai'tan is defeated and Shadowspawn eradicated.. it also explain what happened in the "if world": each soul stolen for a shadowspawn is deprived of rebirth for good except as shadowspawn. This means with each new generation of shadowspawn, more and more animal and human souls are wasted. In the end, human and animal would have no soul left to be born with, leaving only the Shadowspawn to rule... until they wipe each other out.)

The soul borrowed in the making of a Nym isn't tainted, and the soul can be reborn normally afterward. In other words, there are no such thing as an eternal "Nym soul" always born as a Nym. It sounds like among the constructs it's mainly Trollocs that remain close enough to their genetic stock to be able to "grab" an animal soul at conception. IMO, when the human strain dominates, the "trolloc" grabs a human soul, and is born as a throwback, a myddraal, instead. It seems DH can breed, but they need to steal somehow a wolf soul to breed a new DH. Myddraal are sterile, so are gholam. Sammael spoke of the worms as if those were born/made in the AOL, or aren't a stunted breed (I mean, they have not bred true), which can't go through the full biological process planned for those Aginor had made. Zomoran don't breed, and Sammael spoke of cafar as if they were extinct but some might have been put in stasis. We don't know if Draghkar can breed, but it sounds unlikely.

I think the odds are really good that Nym couldn't breed Nym.