Your question is a bad question because induction is impossible and no principle of the uniformity of nature can save it.
Why is induction impossible? Explanations do not follow from observations in any sense. Nor do observations prove any idea. Nor can any observation make any idea one jot more probable. Inductivism is just another variety of justificationism: the idea that it is possible and desirable to prove ideas true or proably true. In reality, you can't prove any position or show it is probable. Any argument requires premises and rules of inference and it doesn't prove (or make probable) those premises or rules of inference. If you're going to say they're self evident then you are acting in a dogmatic manner that will prevent you from spotting some mistakes. If you don't say they are self evident then you would have to prove those premises and rules of inference by another argument that would bring up a similar problem with respect to its premises and rules of inference. Another approach that is sometimes used is that reason has to do with how the human mind works, but since how your mind works doesn't necessarily have anything to do with truth, that doesn't get us anywhere. Justification is impossible, so induction is impossible.
In reality all knowledge is created by conjecture and criticism. You notice a problem with your current ideas, propose solutions, criticise the solutions until only one is left and then find a new problem. Experiments are useful only as criticism. Ideas can't be derived from experiment any more than from any other set of premises. Rather, the idea is that you work out how the consequences of one theory differ from those of another. Then you conjecture ideas about experimental setups that would enable you to see the relevant consequences and criticise them. Once you have a setup that works about as well as you can make it work you use it to do the test. If the results are compatible with one theory and not the others then you may have successfully refuted some false ideas. Sometimes a purported successful experimental test will be successfully criticised because a test is a conjecture about something that happened and that conjecture may be wrong, so experiments don't prove anything.
The idea of the uniformity of nature is totally irrelevant in two respects. First, the mere idea that nature is uniform in some extremely abstract respect like "there are universal laws of physics" doesn't help create specific new ideas to solve problems. It may rule out some bad ideas, but those ideas would be bad by virtue of being bad explanations. For example, the idea that quantum mechanics applies to the whole universe except my house on 22 February 2065 is a bad idea because it introduces an unexplained complication to our ideas about the world that solves nor problems. Second, any such principle would have to be a conjecture about the laws of physics that couldn't be arrived at by induction since you concede that without it induction wouldn't work (it doesn't work with the principle either as I have explained above). So then any such principle would imply that induction is not a complete epistemological theory. This should be looked on as a sign of the bankruptcy of inductivism, rather than being used as an excuse not to dump it in the face of unanswered criticism.
For more on epistemology that is not justified true belief junk, see "Realism and the Aim of Science", Chapter I and "Objective Knowledge" Chapter 1 by Karl Popper as well as "The Beginning of Infinity" by David Deutsch. See also http://fallibleideas.com/.