Hume treats the uniformity of nature as if it were an ordinary inductive principle. What he misses is the fact that experience frequently seems to falsity it and yet we do not necessarily abandon it. When we come across an apparent violation of a generalization uniformity, we have several courses of action short of abandonment, only one of which he recognizes.
We can look for circumstances that differentiate the problematic case, showing that our generalization is not contradicted. (Hume calls this “a secret cause”.) We can modify our generalization by, say, extending its scope. Thus, water does not always boil at 100 Centigrade, but we elaborate the rule to include an additional variable (air pressure). Again, fire does cause burn injuries, but it is not the fire that burns, it is the heat, so cooler things than fire can burn; and acid causes similar injuries. Finally, we can simply park the new case as an unsolved problem - an anomaly, Kuhn describes it. We might abandon the generalization, but only as a last resort.
The Uniformity of Nature tells us when we have a problem and implies how to resolve it. It is not exactly like an axiom or presupposition. It is more like the outline of a research programme.
Newton’s Laws of Motion are equally not inductions. Newton could only have observed a body not affected by any external force in a universe consisting of one body excluding the observer. Nor could he have established or falsified the other laws by observation. In any case, we did not abandon them when we found circumstances in which they do not apply; we simply limited their scope.
These laws are more like axioms, not because they are self-evident, but because they are conceptual (grammatical in Wittgenstein’s sense of the word). They define what a body is. The first one tells what we do not need to explain, and hence what we do need to explain. The other laws are also not inductive. They define a framework of explanation. Lakatos’ idea of a research programme is useful in understanding their function if we regard them as a research programme designed for, and defining, a specific domain.
The principle of the uniformity of nature and Newton’s First Law are similar in some respects, but have different scopes. They don’t assist each other, but then neither needs any assistance from the other.