An eta-product identity for $k=3$, $s=1$ similar to that given in my comment above may be found in "Some eta-identities arising from theta series" by Günter Köhler (Math. Scand. **66**, 1990, p. 146, identity (3)):
$$
\frac{\eta^2(\tau)\eta^2(4\tau)}{\eta(2\tau)}=\sum_{n=1}^\infty\left(\frac n3\right)ne^\frac{2\pi in^2\tau}3,
$$
which is equivalent to
$$
\prod_{n=1}^\infty\frac{(1-q^n)^2(1-q^{4n})^2}{1-q^{2n}}=\sum_{n=1}^\infty\left(\frac n3\right)nq^{\frac{n^2-1}3}=1-2q+4q^5-5q^8+7q^{16}-8q^{21}+...
$$
The rhs is the same as $q^{-\frac13}\sum_{n\in\mathbb Z}(3n+1)q^{\frac13{(3n+1)^2}}$, i. e., up to rescalings, $\frac\partial{\partial z}$ of your $\theta_1(z;\tau)_3$ at $z=0$.

I don't know whether such identities for other $k$ are systematically treated anywhere. Köhler also has a book "Eta Products and Theta Series Identities" but I have never looked inside. From "About this book":

The author brings to the public the large number of identities that have been discovered over the past 20 years, the majority of which have not been published elsewhere.

(Second part)
This might be a separate answer but I decided to just add it here. In "Eta-quotients and theta functions" by Robert J. Lemke Oliver (free online version available), a classification of all those theta-functions of the form $\theta_\psi(z)=\sum_n\psi(n)n^\delta q^{n^2}$ which admit an eta-product decomposition is given. Here $\psi$ is a Dirichlet character, and $\delta=0$ or $1$ according to the parity of $\psi$. There turns out to be very few of this kind - just eight in the even case and five in the odd case. The case of your $k=3$, $s=1$ is the third identity of Theorem 1.1 (2) in that paper; I am not reproducing it here as it is essentially the same as the Köhler's case above. If one allows some twists of the characters, there are a little bit more such products.