Natural language is a very slippery thing; I don't know what one would hope for exactly here. You do clarify that you're looking at "logical natural language," but it's not at all clear what that is. That said, of course one of the roles of logic is to find and analyze well-behaved fragments of natural language, so it's not out of the question that something could be done here - but I still think it's too broad to really hope for too much.
In particular, I would say that hoping to apply a theorem to natural language is pie-in-the-sky. Rather, it's fruitful to look for analogues of constructions in natural language: so, don't try to apply incompleteness to natural language, but rather - motivated by incompleteness - look for an analogue of incompleteness in natural language. This is a subtle distinction but I think it's crucial.
Adopting that stance, we do get something rather interesting when we look at belief and assertion. Consider the following sentence (I believe more-or-less originally due to Hofstadter):
(X) Noah Schweber cannot honestly assert this sentence.
Here of course both "honestly" and "assert" are doing some heavy lifting, but I can confirm that (i) I do recognize that (X) is true but (ii) I can't assert (X) in any way which feels "honest" to me (including point (i) previously - sorry, but honesty on the internet is rare!). Of course, that latter point is subjective - others might experience the situation differently, or argue that there's something silly about invoking qualia here in the first place - but personally I think it's a reasonable point here.
Ultimately of course this leans on a couple assumptions about what's "built into" the relevant part of natural language:
An appropriate mechanism for talking about assertion, belief, honesty, etc.
Classical truth values and behavior, and some capacity for self-reference (at the cost of not being able to refer to truth itself inside a sentence per Tarski).
And an appropriate mechanism for referring to speakers (or "agents").
All of this though is reasonably tame from a mathematical perspective: we can in fact whip up formal "toy models" of this situation. And I would argue that this sentence and its analysis does in fact mirror some of Godel's argument.
But just as interesting as the parallels are the divergences.
An interesting additional feature of (X) is its interaction with universal instantiation. I would say reflexively that I can honestly assert
(Y) For every person P, the sentence "P cannot honestly assert this sentence" is true.
Basically, the point is that I do not consciously percieve all possible instantiations when I consider the universal statement above.
- This recalls a key difference between belief and assertion: one is obviously aware of what one is asserting, but one is arguably not aware of the set of all one's beliefs at any given moment (and even worse, it's reasonable in my opinion to argue that people may have individual beliefs that they don't recognize).
Moreover, it's not clear that there's a habit I can adopt here which is guaranteed to prevent similar problems in the future: I can always make a point to check the specific instantiation "me," but why should that be the only source of problems for universal instantiations here? I'd ultimately argue that this is a point we simply can't avoid:
We can honestly assert universals while not being able to honestly assert - even not believing - some of its instantiations.
There is a family similarity between this situation and the existence on the formal logic side of consistent theories asserting their own inconsistency. However, in my opinion this family similarity doesn't stand up to too much scrutiny. One key point being that in the natural language context this situation is actually happening in the "intended situation," whereas on the mathematical side those self-defeating theories are only true in nonstandard models and in fact only happens as a consequence of a highly-nontrivial technical result (Godel's completeness theorem) whose interaction with natural language is in my opinion even more problematic and interesting (interestingly problematic? problematically interesting?). So there is a real difference in essences here, in my opinion.
I'd say that the final takeaway should be the following:
There are indeed "Godel-like" phenomena in natural language. Moreover, these need not be "cheats" (like "This statement is false" is usually (but not universally) argued to be) - they can coexist with logical tameness and formalizability in other ways. Moreover, further paralleling the mathematical situation at a very general level they lead to further surprises about even "fairly logically tame" natural language; however, these further surprises are not necessarily trivial parallels of ones arising in the classical setting, but should be construed in their own light.