First of all it is not even clear how to <i>state</i> Gödel’s incompleteness theorems in such a weak system, let alone prove it, since the usual statements make reference to computably enumerable axiomatic systems.
One can sidestep that issue by focusing on a specific system of interest. For example you might ask if Robinson’s arithmetic <i>Q</i> can prove Con(<i>Q</i>). You might think that this is just a special case of Gödel’s second theorem, but weirdly enough, that is not quite true, at least not the way the theorem is usually proved, because the arithmetization of syntax is usually carried out in a way that assumes various facts of elementary number theory that in turn assume that exponentiation is total. However, these difficulties can be circumvented, and Bezboruah and Shepherdson did prove <A href="https://doi.org/10.2307/2272251">Gödel’s second incompleteness theorem for <i>Q</i></a>. It is questionable, however, what such a statement really “means” for someone who does not believe that exponentiation is total. The string Con(<i>Q</i>), taken literally, is some kind of complicated algebraic statement, and if you harbor serious doubts about the totality of exponentiation, then it is not clear that this complicated algebraic statement really “expresses” the consistency of <i>Q</i>.
As for exactly what Nelson believed, that was not always easy to ascertain even when he was still alive. Even a weak system of arithmetic admits arbitrarily large numbers, but Nelson said things that indicated that he was suspicious of numbers that cannot actually be written down in unary. <a href="https://mathoverflow.net/questions/142669/illustrating-edward-nelsons-worldview-with-nonstandard-models-of-arithmetic/146523#comment377806_146523">I once asked Nelson</a> about the number $80\cdot 80 \cdots 80$ where we explicitly write down $5000$ copies of $80$ and he was skeptical that this was an actual number, despite the fact that no exponentiation is involved. Under such circumstances, I am not even sure whether Nelson believed that $\sqrt{2}$ is irrational in the same sense that you and I believe that. If Nelson and I were to walk through the proof together, then of course he would agree that every step of the proof was formally correct, but what would the conclusion of the proof “say”? You and I think it says something about arbitrarily large natural numbers but Nelson might not. And if he did not, why should he even believe that the correctness of a short sequence of formal manipulations should tell us anything about (for example) whether a computer search for positive integers $a$ and $b$ such that $a^2 = 2b^2$ would succeed or fail? In short, I do not think it is particularly fruitful to try to understand exactly what Nelson personally believed or doubted, because he simply did not give a sufficiently detailed and coherent account of those beliefs.