From what I've picked up, I think it's quite possible the big American names of the NY Dada scene -- Man Ray in particular comes to mind -- might have lived in obscurity had it not been for more established European artists serving as to anchor the community.
What I mean is, when the European artists went to America, American artists joined Dada as well. If they hadn't, you wouldn't have had a New York Dada but rather a self-contained transplant of the European Dada.
If you have American artists travelling to Europe and being in touch with European artistic circles, they could still pick up an interest in Dada and share it with their colleagues and friends back home. If you can make them get interested in Dada once it got introduced to them, as it happened in OTL, then it's possible that a NY Dada scene would still develop.
As to whether they would have lived in obscurity or had more than a passing interesting, it's hard to say. I think that Duchamp and Picabia's presence as prominent figures of European Dada must have contributed, of course, since they already had solid ties to the Dada scene in Europe. However, if you had any American artists who went to Europe, became at least somewhat important in the Dada circles, and then after going to America still retained their connections to their European counterparts, maybe the work of American Dadaists could still end up being well-known in Europe.