Thanks to Giorgio's comment I found the good reference. In fact De Vore and Lorentz give a refined estimate (Theorem 2.2, Chapter 7) in comparison with the Bramble-Hilbert Lemma I've just cited : $$ \inf_{g\in\mathbf{R}_n[X]} \|f-g\|_p \lesssim \omega_p(f,\frac{1}{n}),$$ where $\omega_p(f,\cdot)$ is the modulus of continuity of $f$ for the $\text{L}^p$ norm. Since any element of $\text{W}^{1,p}(0,1)$ satisfies $\omega_p(f,\delta) \lesssim_f \delta $, so we recover the Bramble-Hilbert cases. In fact, the previous estimate on the modulus of continuity characterizes the corresponding Sobolev space inside $\text{L}^p(0,1)$ (the constant behind $\lesssim_f$ being $\|f'\|_p)$, except for $p=1$, for which this characterizes $\text{BV}(0,1)$ (with constant $\|f'\|_{\text{TV}}$ and not only $\text{W}^{1,1}(0,1)$. In particular, we have therefore, for $f\in\text{W}^{1,1}(0,1)$ $$ \inf_{g\in\mathbf{R}_n[X]} \|f-g\|_1 \lesssim \frac{1}{n}\|f'\|_{\text{TV}}.$$ As I believe the order of convergence $\frac{1}{n}$ is optimal for the Jackson case ($p=\infty$, for Lipschitz functions), this order should also be optimal for the BV class ; indeed, for a Lipschitz function $f$, writing $f=f' \star \mathbf{1}_{\mathbf{R}_{>0}}$, if one is able to a approach $\mathbf{1}_{\mathbf{R}_{>0}}$ in $\text{L}^1$ faster than $\frac{1}{n}$, then Young's inequality would imply a similar superlinear approximation in the Jackson case.

Thanks to Giorgio's comment I found the good reference. In fact De Vore and Lorentz give a refined estimate (Theorem 6.1, Chapter 7) in comparison with the Bramble-Hilbert Lemma I've just cited : $$ \inf_{g\in\mathbf{R}_n[X]} \|f-g\|_p \lesssim \omega_p(f,\frac{1}{n}),$$ where $\omega_p(f,\cdot)$ is the modulus of continuity of $f$ for the $\text{L}^p$ norm. Since any element of $\text{W}^{1,p}(0,1)$ satisfies $\omega_p(f,\delta) \lesssim_f \delta $, so we recover the Bramble-Hilbert cases. In fact, the previous estimate on the modulus of continuity characterizes the corresponding Sobolev space inside $\text{L}^p(0,1)$ (the constant behind $\lesssim_f$ being $\|f'\|_p)$, except for $p=1$, for which this characterizes $\text{BV}(0,1)$ (with constant $\|f'\|_{\text{TV}}$ and not only $\text{W}^{1,1}(0,1)$. In particular, we have therefore, for $f\in\text{W}^{1,1}(0,1)$ $$ \inf_{g\in\mathbf{R}_n[X]} \|f-g\|_1 \lesssim \frac{1}{n}\|f'\|_{\text{TV}}.$$