I can't speak for the American side of this, but in the case of Great Britain, at least, this option was not actually given to the bereaved. A massive scheme of universal interment was devised by the Imperial War Graves Commission under the guidance of Sir Fabian Ware; the scheme ensured that the British fallen would be buried in (more or less) the same earth upon which they died rather than being sent back across the Channel either piecemeal or en masse. I am sure there are individual cases that slipped through the cracks, but in the main this was established as policy.

Our own /u/CrossyNZ has an excellent (and sharply critical) post about the subject here; you may also wish to check out David Crane's marvelous book, Empires of the Dead (2013), which tells this story in greater depth.