A domestic employee in private household, already in the UK holding a purple card can apply for a transfer if employment without the employer needing a work permit (1c).
Changes of employment can be authorised. I've seen the form and the guidance notes so I know it can be done. It's a pain, you have to prove substantial links to the UK and evidence of why you want to continue working in that field / are qualified to do so but in theory it could be done. It's mostly designed for other section 1 workers (airport / minister of religion) but could, in theory, be used for section 1b and c. I never said it was easy, it is merely possible within the scope of the scheme.
If your applicant currently holds a purple card but has been continuously employed for more than 12 months since entry - so is already in the UK and working - they may be freed from the restrictions to work and able to apply for a blue card. Also if they entered the country in that category with a certain passport stamp before 2007 they can remain in that category but switch employer. Or if they're a graduate of a UK institution or have done a postgrad they may qualify for other exemptions...
It's impossible to list all the exceptions so easier to say as a general rule - for a FT nanny - blue card yes, yellow card no, purple card maybe. So ask them what they have an check it.
A blue card, however, is much easier. If they have it you can reasonably suppose they can work. The only thing is, as with any right to work document, to check the name and details carefully and don't be fobbed off with explanations like 'oh they can't spell my name' or 'it's my maiden name'. If there genuinely is a reason for a discrepancy they should be able to back it up.