Remembrance of the Daleks - S25-E1
Corrected entry: We know the episode takes place on 23 November, 1963 because the first episode of Doctor Who is about to come on TV when Ace leaves the room. JFK was assassinated less than 24 hours ago, and nobody seems to be affected. Granted, the military characters have other things on their minds, but none of the civilians wear black armbands or even mention that the biggest news story of the year that just happened.
Remembrance of the Daleks - S25-E1
Corrected entry: The events of this episode take place in Nov 1963, the same time as "An Unearthly Child." However, the schoolchildren are now wearing uniforms and the full name of the school on the sign is Coal Hill Road Shoreditch Secondary School instead of merely Coal Hill School.
Correction: The first episode was broadcast in November 1963, but that does not necessarily mean it was set on that date. For example, the vicar said that the Doctor's voice had changed since he last met him (referring to his regeneration) meaning that the events of Remembrance of the Daleks took place after the First Doctor had left. Othewise, the First Doctor's Tardis would have been in the junkyard. This means that between the events of An Unearthly Child and Remembrance of the Daleks, some time has elapsed which means that the school name & uniforms could have changed.
Remembrance of the Daleks - S25-E1
Corrected entry: When Ace is yelling at Mike for being a stinking dirty scumbag (the second time), she's supposed to be very angry. However, it's plainly clear that she's doing her best not to laugh. Near the end of the scene, she has a grin breaking through, and you can see her actually smirking as she runs off screen.
Correction: Ace was emotionally upset because of Mike, so was probably trying her best not to cry, which can be mistaken as laughter.
Correction: You said it yourself: the people have other things of importance to worry about. This is set in London, where the reaction would not have been as significant as in America. It is set in a school where the lessons would have gone on as normal. At most there would be some discussion in an assembly.