Is there any known uncountable set with an explicit well-order?

There is no known well-order for the reals. Is there a known well-order for any uncountable set?

If not, is it known whether or not an axiom stating that only countable sets can be well-ordered is consistent with ZF?

The backbone of the universe of set theory is the ordinals. We like to forget them, because they rarely come up in analysis or whatever, and even then in order to make certain claims that use these ordinals we often make some use of the axiom of choice.

But the ordinals exist, and pretty much everything that you know about them in $\sf ZFC$ is also true in $\sf ZF$. In particular the existence of uncountable ordinals, which are sets which are transitive, and $\in$ well-orders them. And in the language of set theory, this is as explicit as you can get.

If, however, you remove the axiom of power set, then it is consistent that every set is countable, in which case only countable sets can be well-ordered. But once you have the power set axiom, you can prove the existence of uncountable well-ordered sets, and if you also have the replacement axiom schema, you can find uncountable [von Neumann] ordinals.