Spinor bundle of a fibration over a circle

Question

I have a question: I am reading an article and the author is looking at a submersion $Z \rightarrow M \xrightarrow{\pi} S^1$ with $Z$ is of even dimension. We have $TM = TZ \oplus T S^1$. Given Riemannian metrics on $TZ$ and on $S^1$, we can form the metric on $TM$. We assume that $TZ$ is spin, hence we have the spinor bundle $\mathcal S(TZ)$. On $S^1$, there are 2 spin structures, we choose one and form the spinor bundle $\mathcal S(TS^1)$. It's then not clear to me that $M$ is then spin, and the spinor bundle is isomorphic to $\pi^*\mathcal S(TS^1)\otimes \mathcal S(TZ)$. Could anyone help me with this step please?