Hi Andrew,
1. Have you spoken to your insurance company? Are they willing to provide a temporary vehicle during this time?
2. Just to clarify, it is the other vehicle's insurance company that is informing you that they are only able to provide you with a temporary car during this time? Are they willing to reimburse you the costs for you hiring a rental car if they cannot provide you a car in the meantime?
3. I am a bit confused as to your situation. Is the other vehicle's insurance company only able to provide you with a car while the other driver's vehicle is being repair or while your vehicle is being repairs? Why are they not able to send a replacement car given that i would assume the vehicles would be in the mechanics getting repairs or written off? Are they able to provide any sort of compensation then?