In the end, I had to come to the HN comments because I wasn't 100% sure they meant the homeless; since this is the BBC, and I've seen enough British TV, I know the British themselves use the word homeless.
Maybe this is one of those weird misplaced PC-ification of the language things?
There's countless homeless who aren't sleeping rough on the streets but are taking space, unofficially in someone's spare room, or literally on their sofa or floor. How are the two forms of homelessness distinguished in US usage?
Rough sleeping is as it suggests: Sleeping outside, sleeping bag in shop doorway, tent, abandoned building, under the railway bridge etc.
“Rough sleeping” largely means sleeping outside, or “on the street”.
The rough sleeping set is a subset of the homeless set.
Another commenter suggested that "unsheltered" is the American equivalent to rough sleeping, and I think that makes a lot of sense, even though I haven't heard it much.
Homelessness is obviously an issue in many American cities, and perhaps using more specific definitions could help us understand the issues unique to each group.