After the veto, if Congress does not overturn, what are the options to prevent the national emergency declaration from moving forward?
From all the sources I have seen, the only remaining way to stop a national emergency is to nullify it by proving the president was not acting in accordance with law when doing so. This would have to be done through the courts.
The National Emergency Act contains few words which describe what qualifies as a national emergency. It would be easy to argue this is intentional. However, the US code does provide a definition for "emergency" which would be required:
"Emergency means any occasion or instance for which, in the determination of the President, Federal assistance is needed to supplement State and local efforts and capabilities to save lives and to protect property and public health and safety, or to lessen or avert the threat of a catastrophe in any part of the United States."
-Title 42, Chapter 68, Subchapter I, §5122
Thus it would be left to the courts to decide whether the border situation qualifies as an emergency as defined by our laws. The outcome of such decisions is one of great speculation for both parties.