I was a nanny with my own baby.
I used to fill up their house with travel cots, buggies, toys etc.
But I'd been with them for 5+ years when I had ds. I still look after the youngest after school and all day in the holidays. (at my house now since I've had dd)
Also I used to make sure the house was tidy and all my baby things were as out of sight as possible by the time they came home.
I used a booster highchair that just attached to a normal seat to save space and I brought the travel cot home at weekends.
The children I looked after were all at school by the time ds was born.
I used to bring his own food from home but occasionally he'd have what they were having. I always paid for any enterance fees for him with my own money.
I know lots of nannys with their own babies too.
One looks after 2 girls, they are at school all day. She has a cot etc at their house for her ds.
She does her own thing with him all day then he has to fit in with the mindees after school etc.
The nannies children have to have their naps and classes/groups etc fitted in round the mums job. As long as your nanny does that then it should be ok.
Re the cars, some of them use their own cars which they have bought especially so they are big enough and others have cars bought for them to use by the family. One was given a mini to use with 3 kids. Not very helpfull!
Same with car seats. I have loads of extra boosterseats incase of extra children coming in our car.