Random and Cohen generics over an intermediate model are conull and comeagre

Borel codes are reals, so F1 and ambient AC enumerate all $N$-coded Borel null sets as $(C_n)$. A real is not random over $N$ exactly when it belongs to one of these null sets (every random-algebra dense failure has such a coded null witness). Thus the nonrandom reals lie in $C={\bigcup }_n{C}_n$, which is null by F2.

Similarly enumerate the $N$-coded closed nowhere-dense sets. A real failing Cohen genericity misses an $N$-coded dense open set, hence belongs to its closed nowhere-dense complement. Their union is meagre by F2, so its complement, the $N$-Cohen generics, is comeagre. Empty coded exceptions and a model with finitely many codes are covered by repeating codes in the enumeration.