Many thanks for your detailed reply. I had hoped that after yesterday's hot weather the majority of the bees would have dispersed, but when I checked that evening, they seemed to have bunkered down and were all over the trunk of the tree. This evening, however, despite it being cool and rainy, there has been a marked decrease in the number of them still clustering around. Hopefully by the weekend they'll move on. :)
I've never had experience with a swarm before (it's the first one I've ever seen!), so when the beekeeper said they'd disperse over a couple of days I guess I expected them to up and leave the following day, or stay scattered as they were after he left. The "ball" cluster they formed was unexpected, and what made me wonder if there was a second more juvenile Queen or .. I don't know, something.
To answer your question, the beekeeper didn't leave the box. He was here for about an hour, and after a few unsuccessful attempt to get them to move (because they were inside the pot with the tree) he had to resort to smoking them to calm them, and that was when he got the Queen. Even we could see he'd got her in the box, because it was covered with bees trying to get inside. From our perspective, it was quite interesting to watch, and I'm glad we got them taken away by a beekeeper rather than go with our local council's solution.
I do have a couple more questions, though:
The bees appear to have left behind a white substance on both the tree and the pot - could that be beeswax or the start of honeycomb foundations? It wasn't on the tree before the they arrived.
Is it normal for bees in queenless swarms like this now is to bring back pollen? We've seen a couple of them with rather large amounts of pollen coming back to the cluster, and I didn't think they did that unless they had somewhere to put it?