Maybe my answer is beside the point, because the question speaks about the "limit distribution of $t(\theta_1,\dots,\theta_d)$" (for real $t$?). Shouldn't it be
$n(\theta_1,\dots,\theta_d)$ for $n=1,2,\ldots$? See <https://mathworld.wolfram.com/Kronecker-WeylTheorem.html>. The long answer "from scratch" by 
Peter Humphries proves a different theorem that seems to be more in line with real parameters $t$. Also, the sequence $n(\theta_1,\dots,\theta_d)$ can fill a *disconnected* manifold, which wouldn't properly be called a "subtorus".

If the question is indeed about the *sequence* $n(\theta_1,\dots,\theta_d)$, the book by Kuipers and Niederreiter (Uniform Distribution of Sequences, 1974), contains the generic version of what seems to be the Kronecker-Weyl Theorem as *Example 6.1* on p. 48. The proper condition is the real numbers $1,\theta_1,\dots,\theta_d$ are linearly independent over the rationals.

The notes on p.51 mention that
> a discussion of the exceptional case in this example was also carried out by Weyl,

referring to his classical paper: Hermann Weyl. Über die Gleichverteilung von Zahlen mod. Eins. Mathematische Annalen, 77:313–352, 1916.

Indeed, §5 of that paper, "Die Ausnahmefälle" (the exceptional cases), contains a
*Theorem 18* (pp. 340-341). It deals with the more general case where each coordinate is not just a linear function of $n$ but an arbitrary polynomial.
The conclusion is that the points cover a finite number of affine-linear $r$-dimensional manifolds, (possibly with different integer multiplicities), all these manifolds are parallel, and each of them is filled with uniform density. The theorem specifies how to determine $r$ and the multiplicities.

Here is a statement of this theorem (with slightly different notation), specialized and reformulated for the case of an arithmetic progression $n(\theta_1,\dots,\theta_d)$ as opposed to arbitrary polynomials. The multiplicities are then not necessary.

> Let $\vec\theta=(\theta_1,\dots,\theta_d)$.
Let $S$ be the set of vectors $\vec x\in\mathbb R^d$ such that
$$\langle \vec a,\vec x\rangle\equiv b \pmod 1$$
for all integer vectors $\vec a\in\mathbb Z^d$ and rational numbers $b$ for which the equation
$$\langle \vec a,\vec\theta\rangle= b$$ holds.
Assume that at least one nontrivial equation of this type is fulfilled. Then all numbers $b$ appearing in these equations have a least common denominator $G$.
The sequence $n\vec\theta$ is uniformly distributed modulo 1 in the union of parallel subspaces
$S\cup 2S\cup \dots \cup GS$.  

$G$ is the smallest number $G\ge1$ for which $GS$ contains an integer point (or equivalently, for which $GS$ contains the origin and is therefore equivalent to its linear subspace $S-S$).

Maybe it is an exercise to reduce this case to the independent case?