This is a very interesting question and would make an excellent topic for a doctoral thesis in the history of mathematics.  I will interpret the question as 

*Which pre-Langlands results, problems, and theories* --- *apart from what is easily deducible from the theory of* $\;\mathrm{GL}_1$ (*from Gauß to Tate*) --- *can* now *be considered a part of the Langlands programme ?*

There is nothing original in my answer : everything is gleaned from the writings of Langlands, Serre and Weil.  I may have misrepresented some of their words, and in any case our future doctoral candidate will have to delve deeper into the original sources.

**Fricke** & **Klein** (1912)  observe that the modular curve $X_0(11)$ of level $\Gamma_0(11)$ is defined by the equation $\sigma^2=1-20\tau+56\tau^2-44\tau^3$.  

**Hasse** (193?) asks a doctoral student (Pierre Humbert) to prove that the $L$-function of an elliptic curve $E$ over $\mathbf{Q}$ (defined as the product over various primes $p$ of the $\zeta$-function of $E$ modulo $p$) is entire and satisfies a functional equation.  Humbert sagely decides to work on quadratic forms with Siegel instead.

**Weil** (1951) asks in his report *Sur la théorie du corps de classes* for a galoisian interpretation of the whole idèle class group of a number field (as opposed to the quotient of the said group by the connected component of the identity), analogous to the galoisian interpretation in the function field case.  See http://mathoverflow.net/questions/41318 in this regard.

**Weil** (1952) shows that certain elliptic curves with complex multiplications (such as $y^2=x^4+1$) are modular.

**Deuring** (1953--1957) proves (following a suggestion by Weil) that all elliptic curves with complex multiplications are modular.

**Eichler** (1954) proves that the $L$-function of $X_0(N)$ is essentially the product of Hecke $L$-functions attached to cuspidal eigenforms of weight $2$ and level $N$.  This was generalised by **Shimura** (1958) and completed by **Igusa** (1959).

**Taniyama** (1955) asks at the Tokyo-Nikko conference a somewhat imprecise question which some interpret as implying that one can prove Hasse's conjecture for $E$ by showing that $E$ is modular.

**Shimura** (1966) explicitly determines the reciprocity law for the splitting of rational primes in the number field obtained by adjoining the $l$-torsion ($l$ prime) of the Fricke curve $X_0(11)$ in terms of the coefficient $c_l$ of $q^l$ in the modular form 
$$
q\prod_{n>0}(1-q^n)^2(1-q^{11n})^2
$$ 
(but only for $l<100$ for which he could check that the mod-$l$ representation is surjective).

**Weil** (1967) proves that if an elliptic curve over $\mathbf{Q}$ is modular, then it has to be modular of level equal to its conductor, and assigns the *Übungsaufgabe* to the interested reader to show that every elliptic curve over $\mathbf{Q}$ is indeed modular.

Around this time Langlands wrote a letter to Weil and changed the world.