Mazur's theorem is not needed here, and besides, it applies only to elliptic curves themselves defined over $\mathbb{Q}$ rather than the said compositum $K^{(2)}$ of all quadratic extensions of $\mathbb{Q}$ (which is how I understand the question).

Here is an elementary way to prove finiteness, using height functions and the fact that $K^{(2)}$ has bounded local degrees at each prime $p$. For the last point, simply note indeed that $\mathbb{Q}_p$ has only finitely many quadratic extensions!

Take $p$ a fixed large enough prime, in particular of good reduction. If $f$ bounds the residual degrees of $K^{(2)}$ at $p$, then with $N$ the order of $E(\mathbb{F}_{p^f})$, each point $P \in E(K^{(2)})$ has $[N]P$ reducing to the identity over all primes dividing $p$. Further, if $e$ bounds the ramification indices in $K^{(2)}$ (the remark in the previous paragraph means that $ef$ is bounded!), then writing $\varepsilon_v := [K_v:\mathbb{Q}_v]/[K:\mathbb{Q}]$ the usual normalizing factor for a finite subextension of $K^{(2)}/\mathbb{Q}$ containing $\mathbb{Q}(P)$ and the field of definition of $E$, the contribution to the canonical height of $[N]P$ from the primes dividing $p$ is at least
$$
\sum_{v \mid p} \varepsilon_v \, \lambda_v([N]P) \geq \frac{1}{e}\log{p}
$$
On the other hand, there is a constant $c(E) < \infty$ depending only on $E$ such that, running over all places of $K$, all $Q \in E(K) \setminus \{O\}$ satisfy
$$
\sum_{v \in M_{K}} \varepsilon_v \,  \min(\lambda_v(Q),0) \geq - c(E)
$$
The combination of the two inequalities shows that if $p$ is chosen sufficiently big with respect to $e$ and $c(E)$ then either $[N]P = O$ or it has positive canonical height, hence cannot be torsion. As $N$ is then bounded in terms of $p$ alone, the claimed finiteness of $E(K^{(2)})_{\mathrm{tors}}$ then follows.

The argument works more generally for the compositum of all extensions of $\mathbb{Q}$ of a bounded degree. By Krasner's theorem, such a compositum has bounded local degrees at all places.

A second proof, purely local: Fixing a prime $p$, embed $K^{(2)}$ in a finite extension of $F/\mathbb{Q}_p$. Then your finiteness follows from the well known fact (proved using the formal logarithm and exponential) that an abelian variety over $F$ has finite torsion subgroup.