Explanation
It's not about nesting, it's about the argument. You are doing {a,b,c}[[0]]
.
ResourceFunction["OEISSequence"]["A360142"]
(* {0,1,2,2,4,5,8,9,10,17,18,18,20,35,36,37,40,69,73,74,81,138,145,146,146,148,163,276,291,292,293,296,325,553,582,585,586,593,650,1105,1162,1169,1170,1172,1187,1300,2211,2324,2339,2340,2341,2344,2373,2601} *)
A360142[1]
(* 0 *)
So
A035327[A360142[1]]
is the same as
A035327[0]
Part
zero of any expression is its Head

Therefore,
A035327[0]
(* List *)
As well as
{0,1,2,2,4,5}[[0]]
(* List *)
Solution
ClearAll[A035327,A360142,A035272];
A035272[n_Integer?17>=n>=1] := ResourceFunction["OEISSequence"]["A035272"][[n]]
A360142[n_Integer?54>=n>=1] := ResourceFunction["OEISSequence"]["A360142"][[n]]
A035327[0]
isn't defined! $\endgroup$