The answer of Jørgen Ellegaard Andersen only concerns the case of when the gauge group is $SU(n)$.

I will argue that all the ingredients for the equivalence between the two approaches (namely "geometric quantization of character varieties" and "quantum groups plus skein theory") are out there, for arbitrary simply connected gauge group.

First of all, let us recall what it is that the two approaches actually construct.

$\bullet$ The first approach, developed by Axelrod-DellaPietra-Witten [1] and Hitchin [2]
constructs a bundle over the moduli space of genus $g$ surfaces, along with a projectively flat connection (the so-called bundles of conformal blocks).
Note that there is no way (to my knowledge) to use those bundles to construct 3-manifold invariants without, as an intermediate step, having constructed a modular tensor category.

So I'll take the point of view that "geometric quantization of character varieties" approach only constructs the bundles of conformal blocks.

$\bullet$ The second approach, by Reshetikhin and Turaev [3] takes as input the modular tensor categories coming from quantum groups (aside: the latter were only really sorted out by Sawin [4]). It produces as output a topological modular functor (which assigns vector spaces to topological surfaces with parametrized boundary, and "labels" (objects of the MTC) at each boundary components) and also a 3-dimensional TQFT (a functor from the cobordism category of surfaces and 3-dimensional cobordisms to $Vect$).

As part of the data, we also see vector spaces associated to surfaces, but this time, they are topological surfaces (equipped with a choice of Lagrangian in their first homology).

The vector spaces that appear in the second approach form part of a *topological* modular functor.
A priori, the vector bundles that appear in the first approach are just vector bundles. But actually, they have been shown by Laszlo [5] to agree with the so-called WZW conformal blocks. The latter had in turn been shown by Tsuchiya-Ueno-Yamada [6] to satisfy factorization, i.e., to be a *complex* modular functor.
Given the equivalence between topological modular functors and complex modular functors [7; Theorem 6.7.12], a meaningful question to ask is then:

Is the complex modular functor provided by the first approach equivalent to the complex modular functor associated to the topological modular functor provided by the second approach?

I'll take it that that's the question that the OP wanted to ask.

Andersen and Ueno [8] have showed that complex modular functors are determined by their genus zero data, and the same also holds for topological modular functors. The genus zero data is equivalent (for both complex and topological modular functors) to the data of a `weakly ribbon tensor category' [7, Theorem 5.3.8] (which will turn out to be a modular tensor category in our case of interest).
So we have reduced the question to the following:

Are the modular tensor categories associated to quantum groups equivalent to
the modular tensor categories coming from the WZW modular functor?

The answer is... a complicated yes: see this earlier MO question of mine.

**References**
[1] Axelrod; Della Pietra; Witten,

*Geometric quantization of Chern-Simons gauge theory.* J. Differential Geom. 33 (1991), no. 3, 787–902.

[2] Hitchin,

*Flat connections and geometric quantization.*
Comm. Math. Phys. 131 (1990), no. 2, 347–380.

[3] Reshetikhin; Turaev,

*Invariants of 3-manifolds via link polynomials and quantum groups.* Invent. Math. 103 (1991), no. 3, 547–597.

[4] Sawin,

*Quantum groups at roots of unity and modularity.*
J. Knot Theory Ramifications 15 (2006), no. 10, 1245–1277.

[5] Laszlo,

*Hitchin's and WZW connections are the same.*
J. Differential Geom. 49 (1998), no. 3, 547–576.

[6] Tsuchiya; Ueno; Yamada,

*Conformal field theory on universal family of stable curves with gauge symmetries.* Adv. Stud. Pure Math., 19 (1989), 459–566.

[7] Bakalov; Kirillov,

*Lectures on tensor categories and modular functors.*
University Lecture Series, 21.
(All references are to the

online version.)

[8] Andersen; Ueno,

*Modular functors are determined by their genus zero data.* Quantum Topol. 3 (2012), no. 3-4, 255–291.