If the camera is not disassembled by Mack, it should be OK since a basic cleaning is no more intrusive than if you cleaned it yourself, which Nikon would consider a "user maintenance item". If they get into the cameras, however, Nikon could and likely would void the warranty. Talk to the manager of the store and tell him that he is responsible for any future repair needs during the warranty period of the camera if Nikon voids the warranty.
I have never used Mack but I am wary of any shop that takes in everything from complex pro cameras to hair driers and clock radios. I can't imagine they hire techs who are that specialized and trained, from my own perspective of having a large specialty warranty station for years.
A more immediate problem is that Mack is in limited operation and their database used to show order status is still down, so it would have been faster sending it to Melville which was not down due to the Hurricane. So the store messed up in two ways, sending it to a non-warranty station and sending it to one that is down due to the hurricane, when the proper place for service was not. Have them give you a loaner for your trip, or threaten to reveal their name on a forum with 300,000 members.
In the future, you should consider cleaing yourself. It is not hard or dangerous to the camera. The sensor is not cleaned in a cleaning since it is safely covered up by the AA filter. The AA filter is made of lithium niobate which has a hardness of 5.5 mohs so it is more scratch resistant than tooth enamel or steel. The mirror is made of glass so it even harder. Your chanced of scratching either is low unless you are cleaning in in a dusty diamond mine. Stan St Petersburg Russia