The standard way to show this uses a result from Banach algebra theory (if I can think of a more elementary way to show this, I'll edit my answer to include it).
Let $A$ be a unital Banach algebra with unit $1_A$. If $a\in A$ and $\|a-1_A\|<1$, then $a$ is invertible.
If you haven't seen this, a proof should be found in almost any book on functional analysis, and definitely in one which has a chapter on bounded operators on Banach spaces (or a chapter on Banach algebras).
Returning to the question at hand: since $\|T-U\|<1$, we have
$$\|U^*T-1\|=\|U(U^*T-1)\|=\|T-U\|<1.$$
Thus $U^*T$ is invertible, and therefore $T$ is invertible.