Let $X$ be a quasi-projective smooth complex algebraic surface, with $X^[n]$ the Hilbert scheme of $n$ points on $X$, so that the (rational) cohomology of all these Hilbert schemes together can be generated by the cohomology of $X$ in terms of Nakajima creation operators. Given an algebraic vector bundle $V$ on $X$, there exist universal formulae for the characteristic classes of the associated tautological vector bundles $V^[n]$ on $X^[n]$ in terms of the Nakajima creation operators and the corresponding characteristic classes of $V$. But in general the corresponding coefficients are not known. Based on the derived equivalence of Bridgeland-King-Reid and work of Haiman and Scala, we give an explicit formula in case of the singular Todd classes, but in terms of Nakajima creation operators of the delocalized equivariant cohomology of all $X^n$ with its natural $S_n$-action.