In a situation like this, a legal solution is not always the best.
Moving house is very stressful and the other side obviously got let down at the last moment. It does not sound as if they have done it voluntarily.
Where are you living at the moment? Can you extend your present arrangements easily? If so, I would tend to facilitate the other side.
If it's not inconvenient for you, I would facilitate them. If they need only a week or so, I would not take any legal action. If they are saying it's indefinite, I would issue the Notice of Completion.
If you have a deadline yourself, and can negotiate some compromise with them which suits both parties, you might let them know that you are issuing a Notice of Completion, but just as a backup.
Brendan