President Ronald Reagan’s second inauguration on January 21, 1985, was super cold. The morning low was -4°F and by noon, it had risen to only 7°F. Wind chill values during the afternoon ranged from -10°F to -20°F. Due to this, the public swearing-in ceremony was moved indoors to the Capitol Rotunda, and the inaugural parade was canceled.
Barack Obama’s first inauguration on January 20, 2009, was also fairly cold. Daytime high around 30°F, morning low of 19°F and 28°F at noon during the oath. Wind at 13 mph gusting 23. IIRC There was no snow on Inauguration Day itself, though some had fallen in the days prior and was cleared. .