The only way you have control of the car the nanny selects and drives is to pay for the car. Otherwise your choices are live with it, or get a different nanny. You can demand she use appropriate car seats, again it is up to you to provide the seats.
Perhaps you should let the nanny transport your kids in a car you own. That way you have control of the vehicle and its relative safety.
A '98 BMW 740 is a midsize car and was as good as any other '98 car and likely better than most. If the brakes are in good working order and the suspension and steering are tight it is a “safe” car. Since '98 the major safety improvements have been in more airbags, covering more of the inside of the passenger cabin, and lower pressure airbags.