Evaluate Index of Multidimensional Coordinates
MATL, 8 bytes

PiX[vPi)


This uses 1-based indexing for the node and for the dimensions. So the first nodes are 1, 2 etc; and the "x" dimension is 1, "y" is 2 etc.

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Explanation

The key is to use function X[ (corresponding to ind2sub in Matlab or Octave), which converts a linear index into multidimensional indices. However, the order of dimensions if the opposite as defined in the challenge, so P (flip) is needed before calling the function, and again after concatenating (v) its ouputs.

P    % Implicit input: size as a row vector. Flip
i    % Input: node (linear) index
X[   % Convert from linear index to multidimensional indices. Produces
     % as many outputs as entries there are in the size vector
v    % Concatenate all outputs into a column vector
P    % Flip
i    % Input: dimension
)    % Index: select result for that dimension. Implicitly display