$\newcommand{\al}{\alpha}
\newcommand{\de}{\delta}
\newcommand{\De}{\Delta}
\newcommand{\ep}{\epsilon}
\newcommand{\ga}{\gamma}
\newcommand{\Ga}{\Gamma}
\newcommand{\la}{\lambda}
\newcommand{\Si}{\Sigma}
\newcommand{\thh}{\theta}
\newcommand{\R}{\mathbb{R}}
\newcommand{\F}{\mathcal{F}}
\newcommand{\E}{\operatorname{\mathsf E}} 
\newcommand{\PP}{\operatorname{\mathsf P}}
\newcommand{\ii}[1]{\operatorname{\mathsf I}\{#1\}}$ 

For $i=0,\dots,\lfloor\sqrt{t}\rfloor-1$, we have $t-i=ct$; here and in what follows $c$ denotes various positive quantities (which possibly differ from one another even in the same formula) bounded away from $0$ and $\infty$ over all 
\begin{equation}
\al\in(0,1),\quad T=1,2,\dots,\quad t=1,\dots,T,\quad i=0,\dots,\lfloor\sqrt{t}\rfloor-1;\tag{1}	
\end{equation} 
that is, there is a universal positive real constant $C$ such that for each of these instances of $c$ and for all $\al$, $T$, $t$, and $i$ as in (1) one has $1/C\le c\le C$. So, the sum in question is 
\begin{equation*}
	s_T:=\sum_{t=1}^T (1 - (ct)^{-\alpha})^{c\sqrt{t}}
	=\sum_{t=1}^T \exp\{-ct^{1/2-\al}\}. 
\end{equation*}

If $\al\ge1/2$, then $\exp\{-ct^{1/2-\al}\}=c$ and hence $s_T$ is $cT$ and not $o(T)$. 

If $\al<1/2$, then $\exp\{-ct^{1/2-\al}\}=o(1)$ as $t\to\infty$ and hence $s_T=o(T)$ as $T\to\infty$; in this case, we even have $s_T=O(1)$.