Honestly, there are only two ways to handle this. 1) Consult a suitable reference book. 2) Imagine that what you are writing is going to be published, and look in something published _by that publisher_ that you can use as a model. By "suitable reference book," I'm being evasive. This is the kind of detail that is covered by a "style sheet" or "style book." What book do you use? The answer is this is English and there isn't any Royal Academy and there are lots of them. At one time I personally owned a print copy of the "Chicago Manual of Style," popular _in the U.S._ for academic writing. Yes, this is the kind of detail where U.S. and British usage are likely to differ, and also the kind of detail which evolves with time so you'd better look for something current. To address your particular example, I tried searching theguardian.com, figuring it's a respectable source of good British usage. I used "rowling" first, figuring there would be more hits on the author of the Harry Potter books, and then "tolkien," and found that they used "JK Rowling" and "JRR Tolkien," from which I conclude that this is, at least, acceptable current British usage. If I try the same exercise with the New York Times, I find " J.K. Rowling," period after each initial but no space between the initials. I find BOTH "J. R. R. Tolkien" AND "J.R.R. Tolkien," so what can I do but accuse the New York Times of inconsistency?

It depends. examples of no extra spaces In publishing lots of times, they will just do J.R.R. Token in a name. USA or U.S.A.