Re-typesetting an old paper, what to do with the result?
The copyright will be owned by Nature, so it would be illegitimate (and, as @AnonymousMathematician says, somewhat rude) for you to distribute it yourself.

However sending a re-typeset version to the original authors isn't distribution, but simply correspondence.  If they want to re-distribute this, they're in a better place to do so than you are.  This may not have occurred to them, and your action in suggesting they do so, will be at the very least encouragingly flattering!

I doubt Nature would be opposed in principle.  Their author licence policy says that ‘[Nature Publishing Group] actively supports the self-archiving process, and continues to work with authors, readers, subscribers and site-license holders to develop its policy.’  That's a fairly vague remark, of course, but many universities and funders now mandate 'self-archiving' papers, and journals have no option but to go along with this; therefore the authors may have already have a way of distributing this in a way which is low-hassle to them, and unobjectionable to the journal(s).

Detail: Authors aren't, typically, allowed to distribute the publisher's PDF version of a paper, but are increasingly allowed to 'self-archive' and distribute the 'post-refereeing authors' version' (that is, the same text as in the published version, but typeset by the author).

So, get in touch with the authors.  It sounds like you'll be doing the community a favour.