It is very unlikely that it would be legal for the nanny to be self employed.... (exceptions are where contract are very short (like temp / emergancy nannies/ maternity nurses) or where nannies are working for more than 2 (or 3, not quite sure exactly) families at the same time.
Even if she says she is self employed.... they EMPLOYER can be fined £3000 for not declaring they are employing a nanny... because employers are also legally obliged to pay employers national insurance contributions which you are obviously NOT paying....
anyway that aside you surely cannot seriously be thinking about sacking the nanny because of the car she drives? we nannies are people too, and if we want a sports car or a clapped out old banger (that is insured / MOTed etc) then there is not alot you as employers can do about it...
either get the nanny a car that you are happy for her to ferry your children around in, or make other alternative arangements!
If you do make this girl redunadnt and then employ someone else i woudl be very careful.... if their is no good reason to make her redundant (and im sure in a court her own personal car would not be considered a good reason to sack her!) If you employed someone else (as i assume you would) im sure she could take you to a tribunal.... even if the arrangements change .... hours / pay etc, you are still obliged to offer her the job first...
if she has been with you less than 2 years you dont need to pay redundancy pay, and im sure the minumum notice period is 2 weeks (cause thats whats in my contract) until you have been there 2 years then a weeks notice for every year you have worked up to a maximum of 12 weeks)
The onus is on the employer to provide a contract within 2 months of starting work, so im sure If it came to it everything would be on her side, and then the £3000 fine may come into it as well.
So i dont know what to suggest... be VERY careful about letting her go and then employing someone else... as she could see you in court and would probably win
I personally have my car and with no air bag in the front so i can take 3 children.... i cant fit 3 child seats in the back and will only ever take children in car seats (got no intention of killing / injuring someone elses child just cause a car seat dont fit) have 2 charges at the moment... IF my boss has another baby thats fine.... the option will be never to travel with any of their friends, (maybe leave one at grannies etc) OR they can get me a new car, but I refuse to travel with children not in car seats so that will be it! (and im buisness insured (obvioulsy) and it costs me less than a fiver a year, so it is NOT expensive!)