Natural rights do really require theism to be truly natural, ie, independent of morality and society. Theism avoids the is-ought problem by forgoing the ought, with theism natural law can simply be, and whether you decide you ought to abide them is no longer so important.
Nozick's position on the existence natural rights is simply not grounded. He appeals to intuition and to the reader's morality to appeal for their existence, but he doesn't (and cannot) actually deduce their existence once he forgoes theism. He makes a few appeals to Kant, but they obviously cannot be sufficient, Kant's conditions are merely necessary. I'm very confused by your reference to Nozick on a discussion about the grounding of natural rights when Nozick himself admits that he cannot justify them - he simply assumes Locke, which himself uses a theistic argument, in ASU. If you want, I can get the quote, but I don't have time to skim it until I'm home from work.
At the end of the day, secular natural rights is an intuitive and appealing but ungrounded position that cannot be logically justified, hence why it is threatened by theism. It is no wonder that positive rights and social right theory only really emerged after the Renaissance.