When nanny gets an additional job whilst still in your employ, it is her responsibility to make sure her tax codes in each employment are correct.
As an employer, you could always notify her of her tax situation in your employment and what should happen in her 2nd job. This way you have mitigated any risk of an under payment for nanny and potentially your costs increasing if you have agreed a net wage.
If you are nanny’s main job and have the tax code 1250L for her, if she gets another job, Basic Rate tax (BR) must be applied in her other job. This can be indicated by nanny completing a starter checklist and ticking statement C –