We need the following lemma.
Lemma: Let $C$ be an open convex subset of a Banach space $X$. Let $\mathcal{F}$ be a family of lower semi-continuous convex functions on $C$. If $\mathcal{F}$ is pointwise bounded, then $\mathcal{F}$ is locally equi-Lipschitz and locally equi-bounded.
Let $X',Y'$ be the dual spaces of $X,Y$ respectively. Let $B_{Y'} := \{\varphi \in Y' \mid \|\varphi\|=1\}$. Consider the new collection
$$
\Phi := \{ \varphi \circ T \mid T \in \mathcal T, \varphi \in B_{Y'}\} \subset X'.
$$
We have $| \varphi \circ T (x)| = | \varphi (T (x))| \le \|\varphi \|\cdot \|T(x)\| = \|T(x)\|$. Hence $\Phi$ is pointwise bounded. By our Lemma, $\Phi$ is locally equi-bounded. Then there is $m,r>0$ such that
$$
|\varphi \circ T (x)| \le m \quad \forall x \in B(0, r), T \in \mathcal T, \varphi \in B_{Y'}.
$$
It follows that
$$
\left |\varphi \circ T \left ( \frac{x}{\|x\|} \frac{r}{2} \right) \right| \le m \quad \forall x\in X, T \in \mathcal T, \varphi \in B_{Y'}.
$$
So
$$
\left |\varphi \circ T \left ( x \right) \right| \le \frac{2m}{r} \|x\| \quad \forall x\in X,T \in \mathcal T, \varphi \in B_{Y'}.
$$
Notice that $\|y\| = \max_{\varphi \in B_{Y'}} \varphi(y)$ for all $y\in Y$. So
$$
\left \|T \left ( x \right) \right\| \le \frac{2m}{r} \|x\| \quad \forall x\in X,T \in \mathcal T.
$$
Finally,
$$
\|T\| \le \frac{2m}{r} \quad \forall T \in \mathcal T.
$$
