Associated graded and flatness

Let me suppose, as in your examples, that we have a base field $k$.

It is well known that to check that a right $A$-module $M$ is flat it is enough to show that whenever $Ileq_ell A$ is a left ideal, the map $Motimes_AIto Motimes_A A$ induced by the inclusion $Ito A$ is injective. This condition can be rewritten: $M$ is flat iff for each left ideal $Ileq_ell A$ we have $mathrm{Tor}^A_1(M,A/I)=0$.

So now suppose $A$ and $M$ are (exhaustively, separatedly, increasingly from zero) filtered in such a way that $mathrm{gr}M$ is a flat $mathrm{gr}A$-module.

Pick a left ideal $Ileq_ell A$; notice that the filtration on $A$ induces a filtration on the quotient $A/I$. We can compute $mathrm{Tor}^A_bullet(M,A/I)$ as the homology of the homologically graded complex $$cdotsto Motimes_kA^{otimes_kp}otimes_kA/Ito Motimes_kA^{otimes_k(p-1)}otimes_kA/Itocdots$$ with certain differentials whose formula does not fit in this margin, coming from the bar resolution. Now the filtrations on $M$, on $A$ and on $A/I$ all collaborate to provide a filtration of our complex. We've gotten ourselves a positively homologicaly graded with a canonically bounded below, increasing, exhaustive and separated filtration. The corresponding spectral sequence then converges, and its limit is $mathrm{Tor}^A_bullet(M,A/I)$. Its $E^0$ term is the complex
$$cdotstomathrm{gr}Motimes_kmathrm{gr}A^{otimes_kp}otimes_kmathrm{gr}(A/I)to mathrm{gr}Motimes_kmathrm{gr}A^{otimes_k(p-1)}otimes_kmathrm{gr}(A/I)tocdots$$
with, again, the bar differential, and its homology, which is the $E^1$ page of the spectral sequence, is then precisely $mathrm{Tor}^{mathrm{gr}A}_bullet(mathrm{gr}M,mathrm{gr}(A/I))$. Since we are assuming that $mathrm{gr}M$ is $mathrm{gr}A$-flat, this last $mathrm{Tor}$ vanishes in positive degrees, so the limit of the spectral sequence also vanishes in positive degrees. In particular, $mathrm{Tor}^A_1(M,A/I)=0$.

NB: As Victor observed above in a comment, Bjork's Rings of differential operators proves in its Proposition 3.12 that $mathrm{w.dim}_AMleqmathrm{w.dim}_{mathrm{gr}A}mathrm{gr}M$ (here $mathrm{w.dim}$ is the flat dimension) from which it follows at once that $M$ is flat as soon as $mathrm{gr}M$ is; the argument given is essentialy the same one as mine. I am very suprised about not having found this result in McConnell and Robson's!