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 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.