I'm not on MO often - funny I stumbled on your question...
In general, as you noticed, the analogous statement to Mazur's theorem is false: an example is $\mathbb{Q}[i]$, where the curve $j=1728$ has a rational isogeny of order $p$ for any prime $p=1$ mod $4$.

However, you can still hope for it to hold for fields $K$ which don't admit any CM curves defined over $K$ whose CM field is contained in $K$ (for example this holds for all but finitely many quadratic imaginary $K$ - namely, all except those with class number $1$.)

For $K$ satisfying this hypothesis, I'm pretty sure what you asked is in fact true: there is a finite list of primes p for which a p-isogeny is defined (in fact there's even a finite list of prime powers q with cyclic q-isogenies.) I don't think there's a proof in the literature, but I'm actually writing up a proof of this fact currently with Eric Larson. Our proof uses Galois representations instead of counting points on modular curves and so is very different from Mazur's (and the bounds for p it gives are very big for most fields K). A preprint should be up within the next couple of months.

If you're really interested in a field $K$ that admits such CM curves, you can ask whether there's a finite list of primes that can be isogenies of a *non-CM* curve $E$. This question is harder, since you'd need some way of distinguishing CM from non-CM curves, or at least carefully counting points on $X_0(N)$, and it is certainly still open. However hypothetically this is still true (it falls under the more general hypothesis of Serre that there's a finite list of "exceptional" primes of non-CM curves over a number field K.)