I want to apply the Borel-Cantelli lemma for say events $E_i$ which depend on some parameter $T>0$.
I want a result that says something like: with probability one, only a finite number of $E_i$ is realized. But the thing is I want that number to be independent of $T$.
Let's say I can show that $\sum P(E_i) < \infty $ uniformly in $T$, which using the same arguments as in the proof of Borel-Cantelli, leads to $P(\limsup E_i)=0$ uniformly in $T$.
Is this enough to conclude that with probability one, there exists a finite number (independent of $T$) of $E_i$ that is realized? I feel like it should but don't see why.
Your use of the phrase "the finite number of $E_i$ that is realized (with probability $1$)" suggests that you're reading more into the Borel-Cantelli Lemma than it actually says. It says (when $\sum P(E_i)<\infty$) that the event "only finitely many $E_i$ occur" has probability $1$. It does not say that there is an $N$ such that the event "exactly $N$ of the $E_i$ occur" has probability $1$. And you seem to be asking about just such an $N$. There usually isn't even an $N$ such that the event $B_N$ that "at most $N$ of the $E_i$ occur" has probability $1$. All that one can conclude in general is that $P(B_N)$ approaches $1$ as $N\to\infty$. You might be able to extract information about the rate of that approach from your uniform information about the convergence of $\sum P(E_i)<\infty$, by following the proof of that part of the Borel-Cantelli Lemma.