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 said, there is nothing inherently ineffective in either of the two standard 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 $\mathrm{dist}(\mathbf{k} \cdot \boldsymbol{\theta}, \mathbb{Z}) \geq (n \cdot \| \mathbf{k} \|_{\infty} \cdot H^{D^n})^{-1}$ 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 second of these routes 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, Hlawka). For any sequence $\mathbf{x}_i \in \mathbb{T}^n$ 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 $\mathbf{x}_j := j \, \boldsymbol{\theta}$ and use the Liouville bound as above to estimate the corresponding exponential sums, which are just geometric series having quotients of the form $e( \mathbf{k} \cdot \boldsymbol{\theta} )$ with $\mathbf{k} \neq \mathbf{0}$.