If I understand your question correctly $C_i(X[m])=\oplus_P C_i(X)\otimes U^{i,m}_P\otimes V_P$ where $U^{i,m}_P$ has dimension $d^{i,m}_P$ which is the number of semistandard fillings of the shape $P$ with $0, 1, 2, \ldots, i, \ldots i$ and $V_P$ is an irreducible representation of $S_m$.
For some representations this is easy to compute. For instance if $P_1=m-d-1$ then $d^{d-1,m}_P= 0$, but $d^{d,m}_P> 0$, so $H_{d,P}(X[m])=H_d(X)\otimes U^{d,m}_P\otimes V_P$.