[Luo, Rudnick, and Sarnak][1] prove that if $\pi$ is a cuspidal automorphic representation of $\mathrm{GL}_n(\mathbb{A}_{\mathbb{Q}})$, then
$$
|\alpha_{j,\pi}(p)|\leq p^{\frac{1}{2}-\frac{1}{n^2+1}}
$$
for all $1\leq j\leq n$ and any unramified prime $p$.  The ramification restriction was removed by [Blomer and Brumley][2].

For $1\leq k\leq K$, let $\pi_k$ be a cuspidal automorphic representation of $\mathrm{GL}_{n_k}(\mathbb{A}_{\mathbb{Q}})$.  Consider the isobaric sum $\Pi = \pi_1\boxplus \pi_2 \boxplus \cdots\boxplus \pi_K$.  Then
$$
L(s,\Pi) = \prod_{k=1}^K L(s,\pi_k),
$$
and a bound for the local roots of $\Pi$ at $p$ reduces to a combination of the bounds for each of the cuspidal constituents $\pi_k$.  Here is a convenient and uniform (yet sub-optimal) bound: If $N$ is the maximum of the $n_k$'s, then
$$
|\alpha_{j,\Pi}(p)|\leq p^{\frac{1}{2}-\frac{1}{N^2+1}},\quad 1\leq j\leq n_1+\cdots+n_K.
$$
If $\pi$ (resp. $\pi'$) is a cuspidal automorphic representation of $\mathrm{GL}_m(\mathbb{A}_{\mathbb{Q}})$ (resp. $\mathrm{GL}_{n}(\mathbb{A}_{\mathbb{Q}})$), then one has
$$
|\alpha_{j,j',\pi\times\pi'}(p)|\leq p^{\frac{1}{2}-\frac{1}{m^2+1}+\frac{1}{2}-\frac{1}{n^2+1}}=p^{1-\frac{1}{m^2+1}-\frac{1}{n^2+1}}
$$ where $1\leq j\leq m$ and $1\leq j'\leq n$.

This follows from the explicit description of the local roots (including at the ramified primes) given by Brumley in the appendix to [this paper][3].

ADDED:  In the Selberg class, if you strictly adhere to the axioms, there is no notion of $\alpha_{j,\pi}(p)$.  One has a Dirichlet series for $L(s)$ and for $\log L(s)$, but you are not guaranteed that the Dirichlet series for $L(s)$ factors as a product of local roots like
$$
\displaystyle \prod_{p}\prod_{j=1}^m(1-\alpha_{j,\pi}(p)p^{-s})^{-1}.
$$

  [1]: https://link.springer.com/article/10.1007/BF01895672
  [2]: https://www.cambridge.org/core/journals/canadian-journal-of-mathematics/article/nonvanishing-of-lfunctions-the-ramanujan-conjecture-and-families-of-hecke-characters/90ABA4CD3FEBF0C06BA208EC7036AF3D
  [3]: https://projecteuclid.org/journals/duke-mathematical-journal/volume-168/issue-7/Weak-subconvexity-without-a-Ramanujan-hypothesis/10.1215/00127094-2018-0065.short