Trump can't back down or he undercuts the entire platform he ran on. As the gets more and more painful for me people and the dems continue with "not my problem to fix" attitude the pressure will start to get spread around and people will start asking if it nothing more than pride and Trump hate that is preventing a compromise.
As pressure mounts on democratic senators in areas much more affected by the shutdown then a wall in the middle of nowhere they may pick up enough votes to have bipartisan spending bill that includes the wall. If that happens the whole ugly mess will land in Nancy's lap. Once that happens she has no good options left and will eventually have to piss off her base and deal with Trump. her base hates Trump more than they care about the issue so they will never forgive her for publicly folding.
Her best bet now is to give Trump his damn wall while the pressure is on him and she can get a ton of much needed immigration reform to hell her base salve the butt hurt. Once he has the upper he will use it and use effectively. That is what he is good at.