The numbers $\theta_i$ here are real of course. Also there is only one question really; upon replacing $\boldsymbol{\alpha}$ with $(\boldsymbol{\alpha} - t \boldsymbol{\theta})/s$ the second problem reduces to the case that $s = 1$ and $t = 0$.

That being said, there is nothing inherently ineffective in neither of the two best known proofs of Kronecker's approximation theorem: the diophantine approximations one based on the Dirichlet approximation theorem with an inductive scheme on $n$, and the harmonic analysis one based on Weyl's equidistribution criterion. Into either of these proofs you may input Liouville's lower bound on non-vanishing algebraic quantities, in the form $\mathrm{dist}(\mathbf{k} \cdot \boldsymbol{\theta}, \mathbb{Z}) \geq (2n \cdot \| \mathbf{k} \|_{\infty} \cdot H)^{-D^n}$ for non-zero $\mathbf{k} \in \mathbb{Z}^n$ and $\| \mathbf{k} \|_{\infty} := \max_{i=1}^n |k_i|$.

Let me give some indication of how the latter route goes. It is Weyl's theorem that Erdos and Turan made completely explicit; their inequality is where you were referred to. You will however need the extension of the Erdos-Turan estimate to the higher dimensional torus $\mathbb{T}^n = (\mathbb{R}/\mathbb{Z})^n$, which is due to Koksma. A good reference for this is Theorem 1.21 of the book *Sequences, Discrepancies and Application* by Michael Drmota and Robert Tichy (LNM 1651, 1997). This bounds the discrepancy $D_N := \sup_{I \subset \mathbb{T}^n} \big| \#\{ \mathbf{x}_j \in I, \, j \leq N\} - N \cdot \mathrm{vol}(I) \big|$ of a sequence $\mathbf{x}_j \in \mathbb{T}^n$ by an explicit linear form of its Fourier coefficients:

**Theorem.** (Erdos, Turan, Koksma). *For any sequence $\mathbf{x}_j \in \mathbb{T}^n$ and every $K \in \mathbb{N}$ it holds*
$$
D_N = D_N(\mathbf{x}) \leq N \cdot (3/2)^n \cdot \Big( \frac{2}{K+1} + \sum_{0 < \|\mathbf{k}\| \leq K} \frac{1}{r(\mathbf{k})} \Big| \frac{1}{N} \sum_{j=1}^N e( \mathbf{k} \cdot \mathbf{x}_j ) \Big| \Big),
$$
*with $e(z) := e^{2\pi i z}$ and $r(\mathbf{k}) := \prod_{i=1}^n \max (1,|k_i|)$.*

Apply this theorem taking $\mathbf{x}_j := j \, \boldsymbol{\theta}$. The exponential sums in this case are geometric series with quotients of the form $e( \mathbf{k} \cdot \boldsymbol{\theta} )$ and $\mathbf{k} \neq \mathbf{0}$. They are bounded in magnitude by $(2 \, \mathrm{dist}(\mathbf{k} \cdot \boldsymbol{\theta}, \mathbb{Z}))^{-1}$, which in turn is bounded by the Liouville diophantine estimate above. This is the same point as in the Polya-Vinogradov inequality on character sums.