Earlier quoted context omitted.
But won't replies from other servers show up on the BBC's page if they post something?
Not necessarily, no. There's a whole long standing organization of sysops dedicated to silencing and blocking the (few, I've never even heard of one myself) places on the WWW that the sort of stuff that you are talking about would come from. If the BBC sysops are on the ball, they'll already be looking at what block lists to subscribe to. In fact, they've had the system up since June, and I wouldn't be surprised that…
I was thinking more along the lines of subreddit trolls, who will come in and spam shock/csam images trying to bait the mods, not actual CSAM distributors with known IPs or whatever.
The BBC will certainly attract that kind of internet denizen. They would have to have employees (or automated system) staring at their post replies constantly, watching for this kind of stuff.