Looking in Manhattan, so I'm hoping most schools have a fairly progressive social outlook, but are there any schools that are known for having a capital-c Conservative streak? Would really prefer to not be around a substantial group of Trumpers, for obvious reasons. We're not looking at any uber-religious schools, for what that's worth.

The School at Columbia is very liberal, considering that half of the kids come from Columbia University faculty and the other half comes from the surrounding community making it a very diverse student body and diverse faculty. There is a lot about social justice in the curriculum. They had members of Black Lives Matter come and speak to school before it was a household name.
St Ann's = v progressive
Just so you know, an ultra liberal environment at Dalton has contributed to them not opening. And the liberal administration has told the parents who dissent to be quiet or else. So I’d be careful what you wish for.
Also, look at how they teach health. That says a lot about the schools social outlook.
The upper school student body at Trinity is quite diverse because the 9th grade doubles in size and they draw from all over. While there are new students from the UES and UWS that are demographically more similar to the students who were there in the LS and MS, there are also new kids from downtown, the Bronx, Queens, Brooklyn and NJ. There are kids who come from other independent K-12, K-8, SS, parochial and public schools. There are children of recent immigrants as well legacy families. It's impossible to make generalizations about these students and their parents' social and political leanings since there is a wide range. The school seems to go out of their way to create this wide range in their student demographic.
I'm interested in this question as well( but for HS). How about Trinity? Friends Seminary? Columbia Grammar?
The Blue School
Look at the Upper School English and history curriculum... you can get a sense of how "traditional" the school is based on how they teach these subjects. Some schools stick with a lot of classics that we (and our parents) read many years ago. Others are featuring more modern and diverse authors. Some are a good balance of both. Even within some schools, individual teachers may vary since they have some license to teach based on their own personal interests. This does not necessarily translate into the political leanings of the parent body, but it does provide some context. How they teach history is very telling too. Some schools are really tying in historical teaching with current affairs.
Ethical Culture/Fieldston is the liberal limo of the hill schools and (LS is in Manhattan).
Liberal = Brooklyn private schools