Yes, there exists the result for quantized enveloping algebra. It is developed by Lunts-Rosenberg and proved by them and Tanisaki.
There are several notions:
- quantized flag variety
- quantum D-module.
In the framework of noncommutative algebraic geometry. quantized flag variety is defined as a noncommutative projective scheme. It is a proj-category of quantized enveloping algebra of Lie algebra g. It is a noncommutative separated scheme with affine covers discovered by A.Joseph.
Lunts and Rosenberg defined differential calculus in noncommutative algebraic geometry. They introduced the noncommutative version of Grothendieck differential operators in differential operator on noncommutative ring and then applied this construction to define quantized D-modules in localization for quantum group. In this paper, they formulated the quantized Beilinson Bernstein localization for quantized enveloping algebra in generic case. They proved the global section functor is exact and conjectured it is indeed the correct quantized version, which means it is an equivalent.
Later, under this framework. Tanisaki proved in his paper The Beilinson-Bernstein correspondence for quantized enveloping algebras that the conjecture of Lunts-Rosenberg is indeed true. Moreover, Tanisaki proved this result in root of unity case,see D-modules on quantized flag manifolds at roots of 1.
More comments: In the paper of Lunts-Rosenberg, they pointed out the localization for the quantum group sl2 was constructed by "hand" by T.J.Hodges.
Riemann-Roch for the flag variety is the Weyl Character formula!
More specifically, let $L$ be an ample line bundle on $G/B$, corresponding to the weight $\lambda$. According to Borel-Weil-Bott, $H^0(G/B,L)$ is $V_{\lambda}$, the irreducible representation of $G$ with highest weight $\lambda$, and $H^i(G/B,L)=0$ for $i>0$. So the holomorphic Euler characteristic of $L$ is $\mathrm{dim} \ V_{\lambda}$.
As we will see, computing the holomorphic Euler characteristic of $L$ by Hirzebruch-Riemann-Roch gives the Weyl character formula for $\mathrm{dim} \ V_{\lambda}$.
Notation:
$G$ is a simply-connected semi-simple algebraic group, $B$ a Borel and $T$ the maximal torus in $B$. The corresponding Lie algebras are $\mathfrak{g}$, $\mathfrak{b}$, $\mathfrak{t}$. The Weyl group is $W$, the length function on $W$ is $\ell$ and the positive roots are $\Phi^{+}$. It will simplify many signs later to take $B$ to be a lower Borel, so the weights of $T$ acting on $\mathfrak{b}$ are $\Phi^{-}$.
We will need notations for the following objects:
$$\rho = (1/2) \sum_{\alpha \in \Phi^{+}} \alpha.$$
$$\Delta = \prod_{\alpha \in \Phi^{+}} \alpha.$$
$$\delta = \prod_{\alpha \in \Phi^{+}} (e^{\alpha/2}-e^{-\alpha/2}).$$
They respectively live in $\mathfrak{t}^*$, in the polynomial ring $\mathbb{C}[\mathfrak{t}^*]$ and in the power series ring $\mathbb{C}[[\mathfrak{t}^*]]$.
Geometry of flag varieties
Every line bundle $L$ on $G/B$ can be made $G$-equivariant in a unique way. Writing $x$ for the point $B/B$, the Borel $B$ acts on the fiber $L_x$ by some character of $T$. This is a bijection between line bundles on $G/B$ and characters of $T$. Taking chern classes of line bundles gives classes in $H^2(G/B)$. This extends to an isomorphism $\mathfrak{t}^* \to H^2(G/B, \mathbb{C})$ and a surjection $\mathbb{C}[[\mathfrak{t}^*]] \to H^*(G/B, \mathbb{C})$. We will often abuse notation by identifiying a power series in $\mathbb{C}[[\mathfrak{t}^*]]$ with its image in $H^*(G/B)$.
We will need to know the Chern roots of the cotangent bundle to $G/B$.
Again writing $x$ for the point $B/B$, the Borel $B$ acts on the tangent space $T_x(G/B)$ by the adjoint action of $B$ on $\mathfrak{g}/\mathfrak{b}$. As a $T$-representation, $\mathfrak{g}/\mathfrak{b}$ breaks into a sum of one dimensional representations, with characters the positive roots. We can order these summands to give a $B$-equivariant filtration of $\mathfrak{g}/\mathfrak{b}$ whose quotients are the corresponding characters of $B$. Translating this filtration around $G/B$, we get a filtration on the tangent bundle whose associated graded is the direct sum of line bundles indexed by the positive roots. So the Chern roots of the tangent bundle are $\Phi^{+}$. (The signs in this paragraph would be reversed if $B$ were an upper Borel.)
The Weyl group $W$ acts on $\mathfrak{t}^*$. This extends to an action of $W$ on $H^*(G/B)$. The easiest way to see this is to use the diffeomorphism between $G/B$ and $K/(K \cap T)$, where $K$ is a maximal compact subgroup of $G$; the Weyl group normalizes $K$ and $T$ so it gives an action on $K/(K \cap T)$.
We need the following formula, valid for any $h \in \mathbb{C}[[\mathfrak{t}^*]]$:
$$\int h = \ \mbox{constant term of}\left( (\sum_{w \in W} (-1)^{\ell(w)} w^*h)/\Delta \right). \quad (*)$$
Two comments: on the left hand side, we are considering $h \in H^*(G/B)$ and using the standard notation that $\int$ means "discard all components not in top degree and integrate." On the right hand side, we are working in $\mathbb{C}[[\mathfrak{t}^*]]$, as $\Delta$ is a zero divisor in $H^*(G/B)$.
Sketch of proof of (*): The action of $w$ is orientation reversing or preserving according to the sign of $\ell(w)$. So $\int h = \int (\sum_{w \in W} (-1)^{\ell(w)} w^*h) / |W|$. Since the power series $\sum_{w \in W} (-1)^{\ell(w)} w^*h$ is alternating, it is divisible by $\Delta$ and must be of the form $\Delta(k + (\mbox{higher order terms}))$ for some constant $k$. The higher order terms, multiplied by $\Delta$, all vanish in $H^*(G/B)$, so we have $\int h = k \int \Delta/|W|$. The right hand side of $(*)$ is just $k$.
By the Chern root computation above, the top chern class of the tangent bundle is $\Delta$. So $\int \Delta$ is the (topological) Euler characteristic of $G/B$. The Bruhat decomposition of $G/B$ has one even-dimensional cell for every element of $W$, and no odd cells, so $\int \Delta = |W|$ and we have proved formula $(*)$.
The computation
We now have all the ingredients. Consider an ample line bundle $L$ on $G/B$, corresponding to the weight $\lambda$ of $T$. The Chern character is $e^{\lambda}$. HRR tells us that the holomorphic Euler characteristic of $L$ is
$$\int e^{\lambda} \prod_{\alpha \in \Phi^{+}} \frac{\alpha}{1 - e^{- \alpha}}.$$
Elementary manipulations show that this is
$$\int \frac{ e^{\lambda + \rho} \Delta}{\delta}.$$
Applying $(*)$, and noticing that $\Delta/\delta$ is fixed by $W$, this is
$$\mbox{Constant term of} \left( \frac{1}{\Delta} \frac{\Delta}{\delta} \sum_{w \in W} (-1)^{\ell(w)} w^* e^{\lambda + \rho} \right)= $$
$$\mbox{Constant term of} \left( \frac{\sum_{w \in W} (-1)^{\ell(w)} e^{w(\lambda + \rho)}}{\delta} \right).$$
Let $s_{\lambda}$ be the character of the $G$-irrep with highest weight $\lambda$. By the Weyl character formula, the term in parentheses is $s_{\lambda}$ as an element of $\mathbb{C}[[\mathfrak{t}^*]]$. More precisely, a character is a function on $G$. Restrict to $T$, and pull back by the exponential to get an analytic function on $\mathfrak{t}$. The power series of this function is the expression in parentheses. Taking the constant term means evaluating this character at the origin, so we get $\dim V_{\lambda}$, as desired.
Best Answer
More general setting
The question actually fits into the more general setting that is described in David Ben-Zvi's answer to this MO question (which is about categorification of the Chern character).
HRR for dg algebras
Shklyarov is the one who developped RR theorem for noncommutatve derived schemes (by this one should understand smooth proper DG algebras): http://arxiv.org/abs/0710.1937.
Shklyarov's result has been improved recently by Petit, Lunts, and also Polishchuk-Vaintrob in the context of matrix factorizations.
Let me explain what the statement is. Let $A$ be a proper and homologically smooth dg algebra $A$ (proper means that $\sum_n dim(H^n(A))<+\infty$, and homologicaly smooth means that $A$ is perfect in $D(A\otimes A^{op})$). Let $M$ be a perfect $A$-module. There is a trace map $ch:Hom_{D^{perf}(A)}(M,M)\to HH_0(A)$ (see e.g. this paper of Caldararu-Willerton for a very nice description of the kind of traces I am speaking about), which you can consider as being the Chern character.
Now for an $A$-module $M$ and an $A^{op}$-module $N$ we can consider the $k$-module $N\otimes_AM$ (all my tensor products are derived). Then for $f:M\to M$ and $g:N\to N$ we can consider $ch(g\otimes f)=str(g\otimes f)\in HH_0(k)=k$. Finally the formula is $$ ch(g)\cup ch(f)=ch(g\otimes f) $$ where $\cup:HH_*(A^{op})\otimes HH_*(A)\to HH_*(k)=k$ is the so-called (categorical) Mukai pairing. This is actually more a Lefshetz type formula.
The Todd class is actually hidden in the Mukai pairing (the point is that for associative algebras there is no analogon neither for the Todd class, nor for the usual pairing given by integration).
RR for D-modules
To my knowledge the first one who proved a RR Theorem for D-modules is Laumon (Sur la categorie derivee des D-modules filtres, Algebraic Geometry, M. Raynaud and T. Shioda eds, Lecture Notes in Math. Springer-Verlag 1016 pp. 151–237, 1983). Then Schapira and Scheinders also considered it (Index theorem for elliptic pairs II. Euler class and relative index theorem, Asterisque 224 Soc. Math. France, 1994) and made a very important conjecture which has been proved by Bressler-Nest-Tsygan using methods of deformation quantization ( http://arxiv.org/abs/math/9904121 and http://arxiv.org/abs/math/0002115) developped by Fedosov.
There is also a paper of Engeli and Felder that gives a Lefschetz type formula. Their approach has been later clarified by Ramadoss (he has many paper on this subject that you can find on arXiv).
The subject really moved to deformation quantization. You can learn a lot about all this (with also more details on who one should credit for what) in Section 4, 5 and 6 of this book by Kashiwara and Schapira.
If you have a formal noncommutative deformation $A_\hbar$ of the structure sheaf $\mathcal O_X$ (maybe as a twisted resheaf, or algebroid stack - this is what Kashiwara and Schapira call a DQ algebroid), then you can play the same game as with a smooth proper dg algebras : define a trace with values in Hochschild homology, and state a HRR type Theorem (better, Lefshetz formula) about the compatibility of the cup product on Hochschild homology with the composition of kernels. The reason for that relies on some finiteness and duality properties for cohomologicall complete $A_\hbar$-modules. The main difficulty is then to prove such a result.
To conclude this paragraph, let me observe that (the Rees algebra of) $\mathcal D_X$ can be viewed as a deformation of $\mathcal O_{T^*X}$.
Relation between the question for nc schemes and D-modules
Last but not least, the relation between the derived non-commutative geometry and deformation quantization stuff is adressed in a very recent preprint of Petit: his strategy to prove HRR for DQ algebroids is to use the result for smooth proper DG algebras. Namely, he proves that some derived category of cohomologically complete modules over a DQ-algebroid on a projective variety has a compact generator.
This subject is currently very active.
I apologize for that this answer reduces to a (non-exhaustive) list of references. If you have a more specific question I can tell you where to go in these references in order to (hopefully) find an answer.