Displayed Math in tikz-cd diagrams

You can use \displaystyle\prod and \bigotimes\limits; I also used a shorten length for the curved arrow to prevent overlapping:

\documentclass{article}
\usepackage{tikz-cd}

\begin{document}

\begin{figure}
     \centering
     \begin{tikzcd}[row sep=huge, column sep=huge, text height=1.5ex, text depth=0.25ex]
          \displaystyle\prod_{i=1}^{p} \arrow{r}{i} \arrow[bend right=50,shorten >=10pt]{rr}{T} & \mathcal{M} \arrow{r}{\pi} & \bigotimes\limits_{i=1}^{p}V_i
     \end{tikzcd}
\end{figure}

\end{document}