If there is a yes vote on Thursday, within the next 18 months Scotland will establish its own immigration system and explore its standing in relation to the international rights and obligations of the United Kingdom. For any U.S. immigration policy that is based on reciprocity or a treaty with the United Kingdom, there could be changes applicable to former U.K. citizens/new Scottish citizens.
For instance, for Scottish nationals in the United States in E status, there is a question as to what happens to their visas given that they are based on a treaty with the United Kingdom. The Scottish Government takes the position that international treaties signed by the United Kingdom will continue in effect for Scotland as appropriate (e.g., a treaty with France regarding the Channel Tunnel probably would not continue for Scotland). Another possibility is that Scotland is considered an entirely new state under international law, in which case it would need to negotiate its own treaties unless a specific treaty allows for continuation in this eventuality. Based on my understanding of public international law, the Scottish Government's position sounds plausible, and made more likely by the administrative costs entailed in negotiating treaties anew. I'll be interested to see what the rest of the world has to say if it comes down to it.
Assuming that continuation in effect becomes a reality, the E visa status of former U.K. citizens/new Scottish citizens is unlikely to be affected. This assumes the U.S. government does not object, and so far, I don't believe we have information about the U.S. government's position. If there is an issue with the U.S. government on this point, presumably a new treaty would be negotiated and the current U.K.-U.S. treaty would bridge the gap. In other words, my expectation is that all governments concerned would commit to sensible measures to ensure that no one falls through the cracks.