Hi Lorraine ... what an unfortunate situation to be in at this late stage.
You say that your son in law has arranged an interview under the Premium Service, which is a guaranteed same day service. However, that service is not available for the replacement of lost or stolen passports. Urgent replacement in those circumstances is dealt with under the Fast Track Service, which only guarantees delivery within a week of appointment.
The speed of processing will not just depend on how much pressure the IPS are under at the moment, it may also depend on the accuracy of the application and supporting documentation, which will enable them to verify his status and eligibility. Your son in law will need to ensure that all paperwork and photos are completely in order, if he is to avoid a wasted appointment and a delay to his application.
It may be worth looking at the terms of your travel insurance, to see if any cover is provided for additional expenses or booking alterations related to the loss or theft of a passport before departure.
If there is any suspicion that the passport has been stolen, it should also be reported to the police.
Good luck, please keep us informed.
David