Car Appraisal: Insurance makes problems

Hello, I had an accident, an about 90-year-old lady drove me in, who could hardly hold herself on her feet. She got out and greeted me with the words “Was that me or was that already like that before”. Police called only very, very reluctantly (O-Sound: we have more important things to do). But since the lady appeared very sensibly I insisted on it. They took the damage only briefly and gave in a note, put no file number or the like. Car in the workshop worked. The workshop has now independently ordered an expert who has prepared an expert report without my order. This consists of 9 pages and consists only of pictures of the vehicle. The damage is estimated at 2,350 € (the damage is in my opinion much less, but doesn’t matter) and the expert has now created an invoice of 650 € (with what hourly wages do they work with? 300 € the hour?). Now the invoice should be paid immediately, but I did not have I always thought that this was the case with the generic insurance. Quotation: With regard to the assumption of the expert costs applies: In case of an accident without fault, the causer of the accident or his liability insurance takes over the costs of the car expert. Source: Now I have turned to the opposing insurance which mine but do not pay that, it would not be all 100% clarified. I don’t know what they want to clarify yet. There are unfortunately no witnesses for the accident from both sides. Besides, I did not order the expert at all but the workshop itself: can I take it somehow in regress? Alternatively, I try to talk to the expert that the bill will be paid if the opposing insurance means everything would be settled. Let them talk to themselves? How long does it usually take to process such a case with an insurance company & be fully paid out? what about months? :/ Greetings!