You are expected to cover your own travel costs to and from work (this typically involves travelling to the child / children's school or family home). However, any travel costs incurred during your shift (for example, bringing a child / children home from their school to the family home on a bus) will be reimbursed by the family.
You should discuss what your family deem ‘allowable’ expenses at the start of your employment. For instance, some families may say it’s OK to take the kids home in taxi if there’s a tube strike, while others might expect you to use the bus - it’s important you get the permission in advance, otherwise your expense might not be reimbursed.
It's also good to clarify a method of claiming expenses upfront to avoid any confusion. Some families arrange reimbursement directly (i.e. provide the nanny with an Oyster card they top up) while some nannies report expenses at the end of the month and get paid that way.