= Solution
The set $T$ is uncountable. Partition $\mathbb N$ into countably many infinite sets $B_1,B_2,\ldots$, and choose bijections $h_j:B_j\to\mathbb Z$. For every subset $A\subseteq\mathbb N$, define a permutation $f_A$ separately on each block by
$$
h_j(f_A(x))=
\begin{cases}
h_j(x)+1,&j\in A,\\
h_j(x)-1,&j\notin A.
\end{cases}
$$
Every positive iterate translates each integer coordinate by a nonzero amount, so it has no fixed point and $f_A\in T$. Distinct subsets $A$ give distinct permutations, yielding an injection $\mathcal P(\mathbb N)\hookrightarrow T$. Since the <power set> is uncountable,
$$
\boxed{T\text{ is uncountable}}.
$$
In fact its cardinality is the continuum, because it is a subset of all functions $\mathbb N\to\mathbb N$.
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