@WillSawin thanks for keeping me on my toes ;).

Nice! I did state in my "answer" that I assumed the curve was defined over $\mathbb{Q}$, and in fact as you'll see from my comment on the accepted answer, I assumed (based on the fact that that answer was accepted) that OP had intended that. If OP wanted to know what torsion could be defined over $\mathbb{Q}^{(2)}$ for an elliptic curve which is just know to be defined over that field, I don't think we know the answer, although your argument apparently shows the group is still finite in this case.

@Suman First of all, I assume since you accepted this answer that you meant in your original question that the elliptic curves should be defined over $\mathbb{Q}$, and you want to know about $E(K)_tors$ for such curves, where $K$ is the compositum of all quadratic fields. The fact that the curves are defined over $\mathbb{Q}$ is of course crucial in the Fujita & Laska-Lorenz results mentioned above. Secondly, I have an easy way to see that the order of the group is finite. It's maybe too long for a comment here, so see my "answer" below.

I see -- you mean isomorphic as Galois modules.

@JSE You mean $\mathbb{Q}(E[p])$ isomorphic to $\mathbb{Q}(F[p])$ implies $E$ and $F$ are isogenous, right? Do you have a good reference for this conjecture?

I am guessing it should say "...degree less than $D-l$," not "...$D-d'$."

Thank you very much for bringing this argument fact (re Conjecture 2'') to my attention! I guess this is the reason for the focus in recent literature on conjectures 1, 1', etc. In your opinion, is there a good reason not to believe in Conjecture 1''? It would be nice if one could say all bounds in the OIT only depend on $[K:\mathbb{Q}]$.