Let $K$ be a quadratic field, and $E/K$ a non-CM elliptic curve with a $K$-rational $p$-isogeny, for $p$ a prime. I would like to say the following:

For large enough $p$, the $j$-invariant $j(E)$ must be in $O_K$.

I would also like to know exactly how large $p$ must be. (This lower bound may very well not depend on $K$.)

The above is indeed true when $K$ is replaced with $\mathbb{Q}$; 37 is definitely large enough; (though the true bound may be as small as 17. EDIT: Actually, it's 19. See the comments.)

Corollary 4.3 in Mazur's article [1] gives me hope that the above may be true. Assuming condition A (which I'll describe presently), it says (for my set-up) that the only primes which can divide the denominator of the $j$-invariant are the primes above 2 and 3; (moreover, if 3 doesn't ramify in $K$, then it's only the primes above 2 that need concern us).

Condition A is that $J_0(p)$, the jacobian of the modular curve $X_0(p)$, possesses an "optimal quotient" whose Mordell-Weil rank over $K$ is zero.

So I guess I'm hoping for two things:

That these small primes can be dealt with (i.e. for $p$ large enough, they don't arise in the denominator of $j$); and

Condition A can be removed after all of these years (at least for $p$ large enough).

EDIT (after comments from Felipe Voloch and Noam Elkies): Merel's "winding quotient" has rank 0 over $\mathbb{Q}$; but I don't think it will have rank zero over every quadratic field. I also don't know over which quadratic fields the winding quotient has rank zero.

Noam Elkies is quite right that the desired result is "vacuously true", since it is known (by work of Momose, Theorem B in [2]) that there are only finitely many $p$ for which there is a $K$-rational $p$-isogeny. However, I'd still like a more direct approach to the lower bound question, not using Momose's much stronger result, if there is one…

[1]: Mazur, B. "Rational Isogenies of Prime Degree". Inventiones, 1978.

[2]: Momose, F. "Isogenies of Prime Degree over Number Fields". Compositio, 1995