@katliamom, though what the prestigious universities abroad consider top American universities may differ some from what Americans may think.
If I have to generalize, I’d say they tend to respect the big research heavyweights.
As an example, UIUC has exchange programs with UTokyo, Keio, and Waseda (the top uni in Japan and the top 2 private unis in Japan, which is why Keio and Waseda are sometimes called the Harvard and Yale of Japan) and has a ton of study-abroad programs in general.
I notice that many of the top American privates tend to organize their own study-abroad programs as a consortium (for example, the Kyoto consortium organized by Columbia, the EDUCO consortium in Paris, etc.)
With 3-year undergraduate programs, LSE (and Oxbridge) don’t seem to do formal undergraduate exchanges with almost any American schools (though other English unis do). But there are ways to study at LSE and some of the Oxbridge colleges, and as you said, LSE offers a 1-year study abroad course (http://talk.collegeconfidential.com/study-abroad/1047994-lse-study-abroad-p1.html)