There's a beautiful history behind this.
Basically, Artin and Hecke were working on different sides of this "dihedral modularity conjecture" at the same time (the 20s) and at the same place (Hamburg), but apparently they never discussed this aspect of their research.
So they had the tools to prove this instance of what would become the Langlands program by 1927, but they didn't know it!
There is a brief account of this in Tate's paper "The general reciprocity law" (note that he was Artin's doctoral student), and a more extended historical survey of Artin and Hecke's work during that time on Cogdell's article "On Artin L-functions".
I think this is how the proof would have looked like back in 1927 (although in modern notation, and not in german!)
Arithmetic side (Artin)
Let $\rho:\mathrm{Gal}(L/K)\to \mathrm{GL}_2(\mathbb{C})$ be 2-dimensional dihedral complex representation. From representation theory we know that $\rho$ is monomial, that is, induced from a 1-dimensional representation.
Artin had proved in 1923 that his L-functions behave well under representation theoretic operations and, in particular, induction. Therefore, there is an L-function $L(\varrho,s)=L(\rho,s)$, with $\varrho$ one-dimensional (abelian).
From Artin reciprocity (1927) we have that $L(\varrho,s)=L(\chi,s)$, with $L(\chi,s)$ a Hecke L-function.
The last step is Hecke's proof from 1917 that abelian L-functions are meromorphic for non-trivial characters. Since $\varrho \neq 1$, the original L-function $L(\rho,s)$ is meromorphic on the complex plane, and we have proved the Artin conjecture for dihedral representations.
Automorphic side (Hecke)
Hecke had been studying theta series, and in particular in 1927 he constructed a cusp form $f_\theta$ of weight $1$ as a linear combination of $\theta$-series of binary quadratic forms attached to $K$.
He had alredy proved the basic properties of the L-functions of arbitrary modular forms and Hecke characters, so he knew their functional equation.
In the case of his $f_\theta$ the gamma-factor was very simple, just $\Gamma(s)$.
So, according to Tate, he listed all the Hecke L-functions that shared that same gamma-factor. After weeding out the one coming from Eisenstein series (which in turn correspond to cylic (reducible) two-dimensional representations), he was left with a correspondence $L(\chi,s)=L(f_\theta,s)$.
Arithmetic side revisited
This would have been an easy step for either one of them, if they had known what the other one was up to.
A quick inspection of the gamma-factor of the Artin L-function shows that the only representations for which it equals $\Gamma(s)$ are the ones odd and two-dimensional. Since the other two-dimensional odd representations are irreducible (except the cyclic, which we have alredy mentioned correspond to Eisenstein series), we have showed:
$$L(\rho,s)=L(f_\theta,s)$$
Jacquet-Langlands proof
The first actual proof of the result follows from the converse theorem for $\mathrm{GL}_2$ in "Automorphic forms on GL(2)" (1971). But I don't think they mention the dihedral case in particular. Langlands does, saying that it is implicit in the works of Hecke and Maass, in his 1975 book "Base change for GL(2)".
A different proof follows from the results by Deligne and Serre in "Formes modulaires de poids 1" (1974).
I'm not sure of what relevance Maass' work has in this case. The same goes for some attributions to Brauer, since his induction theorem isn't really needed here.
To answer the actual question, no, there's no direct reference for this result before 1971. That said, technically Artin's 1927 paper implies this case of the (weak) Artin conjecture, and we now know (by a result of Booker, 2003) that this "weak" case implies the strong Artin conjecture.
Several remarks before answering your questions: (1) Langlands-Tunnell is a result in the other direction: from Galois representation to automorphic forms; it is therefore not relevant. (2) One expects to be able to attach Galois representations only to
certain types of Mass forms, those whose component at infinity in algebraic (in the automorphic representation settings) or equivalently, whose eigenvalue for the Laplacian is $1/4$. (3) this Galois representation is expected to take values in Gl${}_2(\mathbb{C})$, hence to have finite image.
So you ask: "how does one construct a galois representation from a Maass form?".
The answer is: one still doesn't know how to. It's one of the most striking open problem in the Langlands program. There was 25 years ago an announcement that this problem
has been essentially solved (with published articles), but it was soon after retracted:
see the two references given by Chandan in comments.
And for your displayed question about the infinite family, stripped of the reference to Langlands-Tunell and of the "infinite image" condition, the answer is no, as far as I know,
Best Answer
To lift $\bar{\rho}$ to a geometric representation in the sense of Fontaine-Mazur, the standard technique requires that $\bar{\rho}$ is balanced, i.e. the dimension of a certain Selmer group must equal the dimension of its dual Selmer group (associated to a certain deformation problem). This is not the case for even representations. However, if one relaxes the $p$-adic Hodge theoretic condition at $p$ (crystalline or de Rham for example) then it becomes possible to sometimes lift an even $\bar{\rho}$ to a characteristic-zero representation $\rho:G_{\mathbb{Q}}\rightarrow \text{GL}_2(\mathbb{Z}_p)$ which is ramified at only finitely many primes as was demonstrated by Ramakrishna in "Deforming an Even Galois reprentation". In this paper, some even representations were lifted to characteristic zero for $p=3$. These lifts should not satisfy the de Rham condition at $p$. On the other hand, you're not expected to get geometric lifts even for small primes ($p>7$ was deduced in Calegari's paper).
If you're looking for lifts which satisfy a local condition at $p$ you will get lifts which are ramified at an infinite density zero set of primes. These may be constructed through an application of the lifting strategy of Khare, Larsen and Ramakrishna. You may be interested in knowing where such representations in general come from.
Also, it is worth noting that the standard geometric lifting technique does not apply for residual representations $\bar{\rho}:G_{K}\rightarrow \text{GL}_2(\mathbb{F}_{p^m})$ if $K$ is not totally real since in this case the associated Selmer and dual Selmer groups do not match up in dimension. This in no means implies that there are no geometric lifts when $K$ is imaginary quadratic (for instance).