Just reporting back here. I've now bought two of these boxes.

#1 in mid June was super thick, quite long, quite high, but had no handles. On the plus side, it had "bicycle box" written all over it. The size meant it was very easy to pack (no need to remove rear wheel, pedals, seat or front rack). On the downside, it was pretty tough to lug around, especially without handles - ended up carving a few in the side, and also making some out of packing tape. I called up on the morning to reserve one at Terminal 4 - they seemed happy enough to make sure there were boxes at whatever terminal you need.
#2 , in early July was completely different - about the normal size, with handles, but without any printing on the outside. The guy mentioned something about a different supplier. We tried to reserve, but this guy said there was no need.

We spent a while trying to work out a better solution than handing over the 25 pounds per box. Couldn't come up with anything that was guaranteed, less stressful, significantly cheaper and didn't require awkward steps like trying to transport bike and box together but unboxed.

Also, if anyone's wondering, to get to Heathrow Terminal 4 from Reading, we took the airport bus - they weren't thrilled about taking unboxed bikes, but said it was ok since there wasn't much other luggage. They've had lots of issues with bikes destroying other people's luggage, apparently. It drops you at Terminal 5, and it's a comfortable and safe ride from there to Terminal 4 - bike path most of the way, then a dodgy footpath for the last 500m.