Critair only mandatory in some cities, not nationwide, so it will depend on where you plan to drive
https://www.rac.co.uk/drive/news/mot...french-cities/

I'm not sure about delays and how long you'll keep the car, but a critair sticker costs about 4 € so might be worth applying for it as soon as you get the car if indeed you are traveling into one of the "Critair cities"?