What new house? Is this the house you are buying?
If your buyer has signed unconditional contracts, then they must complete. If they have specified some conditions, then you must meet those conditions.
If they do not complete, you can keep the deposit and sue them to complete, although this is usually not worthwhile.
If you are worried, why not sign the contracts yourselves and send them back and move out as soon as possible?
You may have to find temporary accommodation or you may agree with them an extended closing date.