ZF proves that whenever a countable union of countable sets can be well ordered then its cardinality is at most $\aleph_1$. But what if it cannot be well ordered? The FefermanLevy model shows the continuum can be a countable union of countable sets. Is there a ZF proof that a countable union of countable sets must be smaller than or equal to that in size?

In Jech The Axiom of Choice, problem 14 on chapter 5 states:
Note that $\mathcal P(X)$ can only be mapped onto set many ordinals, so this ensures us that there is indeed a proper class of cardinalities which can be written as countable unions of countable sets. Jech adds and points out that this $N$ is not an inner model of any transitive model of ZFC, as that would imply $2^{\aleph_0}$ can be partitioned into any number of sets. The reference given is to Morris from 1970:
I haven't sat down to verify all the details, but it should probably work: Using an Easton symmetrylike class forcing, we may obtain model add for every regular $\kappa$ a countable collection of countable subsets of $2^\kappa$ whose union is not countable, let us denote this union $A_\kappa$. Genericity arguments should give us that for $\kappa\neq\kappa'$, $A_\kappa$ is incomparable with $A_{\kappa'}$  namely there are no injections between those two sets. This shows that there is a proper class of different sets which can be represented as countable unions of countable sets. So no upperbound is possible. 

