Everybody in town’s father was playing, it seemed, except Atticus.
is in a colloquial register which I believe is suitable for the tone of the narrative.
In both of your possible variants, the subject of the sentence is an entire phrase which could be misconstrued (though it would be nonsensical).
{Everybody in town’s father} was playing, it seemed, except Atticus.
could be misconstrued as
Everybody {in town’s father} was playing, it seemed, except Atticus.
(There exists an entity "town's father" which everybody is at/inside.)
Similarly,
{Everybody's father in the town} was playing, it seemed, except Atticus.
could be misconstrued as
{Everybody's father} in the town was playing, it seemed, except Atticus.
(The whole town has the same father.)
The most unambiguous wording I can think of is
The fathers of everybody in town were playing, it seemed, except Atticus.
[T]he problem we confront when creating possessives with compound constructions such as daughter-in-law and friend of mine. Generally, the apostrophe -s is simply added to the end of the compound structure: my daughter-in-law's car, a friend of mine's car. If this sounds clumsy [or is ambiguous], use the "of" construction to avoid the apostrophe: the car of a friend of mine, etc
– Edwin Ashworth Dec 27 '14 at 13:10