I believe that $\lambda_1$ is of order $T^{-2}$.  One can get the upper bound by a direct analysis of the Rayleigh quotient:  the function
$$
f(x)=\begin{cases}\sin(\pi t/T),& \text{for $x=(s,t)$ in the neck part $S\times [-T,T]$}\\0,& \text{for $x$ not in the neck part}
\end{cases}
$$
is in the function space $H^1(M)$ and $\int_Mf=0$, so
$$
\lambda_1\leq \frac{\int_M|df|^2}{\int_Mf^2}=\frac{\pi^2}{T^2}.
$$
For the lower bound, as noted by @Neal, it seems that the Cheeger constant $h(M)$ should be achieved by a cross-section $S$ in the neck part, so
$$h(M)=\frac{\operatorname{Vol}(S)}{\operatorname{Vol}(M)/2}
=\frac{\operatorname{Vol}(S)}{T\operatorname{Vol}(S)+c}\sim \frac{1}{T}.
$$
This means that $\lambda_1\geq h(M)^2/4\sim\tfrac{1}{4}T^{-2}$.