This paper establishes a comprehensive well-posedness and regularity theory for time-fractional stochastic partial differential equations on $\mathbb{R}^d$ driven by mixed Wiener--Lévy noises. The equations feature a Caputo time derivative $\partial_t^α$ ($0<α<1$) and a spatial nonlocal operator $φ(Δ)$ generated by a subordinate Brownian motion, leading to a doubly nonlocal structure. For the case $p \ge 2$, we prove the existence, uniqueness, and sharp Sobolev regularity of weak solutions in the scale of $φ$-Sobolev spaces $\mathcal{H}_p^{φ,γ+2}(T)$. Our approach combines harmonic analysis techniques (Fefferman--Stein theorem, Littlewood--Paley theory) with stochastic analysis to handle the combined Wiener and Lévy noise terms. In the special case of cylindrical Wiener noise, a dimensional constraint $d < 2κ_0\bigl(2 - (2σ_2 - 2/p)_+/α\bigr)$ is obtained.~For the low-regularity case $1 \le p \le 2$, where maximal function estimates fail, we construct unique local mild solutions in $L_p(\mathbb{R}^d)$ for equations driven by pure-jump Lévy space-time white noise, using stochastic truncation and fixed-point arguments. The results unify and extend previous theories by simultaneously incorporating time-space nonlocality and jump-type randomness.