Short answer: The estimate is similar to that for functions in one variable.

Longer answer: The estimates of best approximation of a real-valued smooth function (by algebraic as well as by trigonometric polynomials) in terms of moduli of continuity of its derivatives are known as Jackson theorems. They were first proved for functions on the unit interval by Dunham Jackson. For functions on the $n$-cube a result of this type was obtained by D. J. Newman and H. S. Shapiro [On approximation theory (Oberwolfach, 1963), pp. 208–219, Birkhäuser, Basel, 1964; MR0182828 (32 #310)]. I cannot give now an exact quotation, because I do not remember the result well and do not have an access to the paper. There were subsequent generalizations, many by M. Gansburg. One of the newer results is the following theorem from MR1920286
Bagby, T.; Bos, L.; Levenberg, N.;
Multivariate simultaneous approximation.
Constr. Approx. 18 (2002), no. 4, 569–577:

If K is a connected, compact set in $\mathbb{R}^N$ such that every pair of distinct points $a,b$ of $K$ can be joined by a rectifiable arc in $K$ with length at most $s|a-b|$, where $s$ is a positive constant, and $f$ is a function of class $\mathcal{C}^r$ on an open neighborhood of $K$, then for each non-negative integer $n$, there exists a polynomial $p_n$ of degree at most $n$ defined on $\mathbb{R}^N$ such that for each multi-index $\alpha$ with $|\alpha|=\min\{r,n\}$,
$$\sup_K |D^{\alpha}(f-p_n)|\leq \frac{C}{n^{r-|\alpha|}}\sum_{|\gamma|\leq r}\sup_K|D^{\gamma}f|,$$
where $C$ is a positive constant depending only on $N$, $r$ and $K$.