OP it depends what kind of CPD you think your nanny would be interested in doing. MNT do good courses but focused more on young babies. There's a nutrition course but it's London based. The OU do courses but that's quite a commitment.
Outraged Questions questions 
'frak do you think your process is representative of the average nanny employer (if there is such a thing!)? I can't imagine any of my employers have ever looked that closely at applicants qualifications. Knowing a lot of the mums who have nannies, I highly doubt they did. Given some of the 'haven't a clue' questions we get on here, I doubt they've gone into checking who has accredited their nanny's qualifications.'
Of the average employer, probably not, but it should be close to what an agency or someone who has worked in the sector or HR does. Most people are bogged down in the major qualification - level 3, CACHE/BTEC/NVQ, early years/childcare and education/CCLD, whatever else that the CPD barely gets a look in. Seeing as I know at least the major qualification titles, awarding bodies and dates when they were offered by heart I don't really stress about that. It gives me more time to delve. Best in mind as an employer I have been sent a handful of CVs with any CPD on at all, and I'm including the MNT maternity practitioner and training done in nurseries in that. I suspect ex-nursery nannies have things they aren't declaring.
'Out of interest what is your approach when you receive job applications. Do you go through and chuck any unqualified nannies in the bin, then sort through the rest in order of who has the most courses? Do covering letters count for anything? Would you ever interview an unqualified nanny? Or a nanny who hasn't done anything since originally qualifying? Do you think gut instinct is important at all or is it a 'who looks best on paper' deal?'
First I check my non-negotiables (EU passport, French or English MT level, availability etc) then my desirables (age appropriate experience, qualification, CPD, good GCSEs etc) and if they don't tick all essentials and at least 50% of the desirables they don't go into the 'priority pile'. Some are an either/or, particularly relating to qualifications/experience.I do read covering letters if attached, as an employer I've rarely had them sent to me. It's usually a brief email along the lines of 'I'm interested in your job this is my CV'. Gut instinct plays at interview, it doesn't help me decide who to interview. Courses do not a good nanny make - the worst nanny we interviewed was from a highly regarded trainjng college but had no warmth or communication skills - but someone who qualified 20 years ago and has done no training since isn't, IMO, helping themselves. Experience doesn't speak for itself, a lot more is asked of nannies educationally these days and education, particularly early years, is quite a fast moving field. I would rather see experience plus short courses than an NMEB who has done nothing since. I have interviewed nannies with less than 5 years experience who haven't done anything since which I feel is acceptable, just about. I have also interviewed a totally unqualified nannies (nothing after GCSE) or nannies without childcare qualifications but related higher educational qualifications (psychology degree etc). That makes me sound like I've interviewed hundreds of nannies to work for me. I haven't, I've interviewed maybe 10 British nannies? From the 2 nannies I've employed 1 had done CPD when working in a nursery and 1 hadn't specifically done CPD courses but read a lot. That came out entirely accidentally, it wasn't anywhere on her CV and I nearly didn't interview her, but I mentioned on the phone we loosely follow attachment parenting and she burst out 'I've read about that yadda yadda yadda' which demonstrated that she actually had, and not just a sensationalist article online. (And yes I do phone and then in person interviews, it's easier than interminable email exchanges).
'I get the bit about it looking good on a CV, my question is do they actually improve the quality of care? Do you think if you had video footage/could observe nannies with their charges or read references or something you'd be able to pick out the ones with 'extra courses' against the ones without? What differences would you expect to see?'
Based on other early years settings I would expect childcarers to be able to use it in their work. For example if they've done a specific training course on schema I'd expect them to be able to identify and naturally extend a schema presented by a child. Someone without that training may not be able to a) identify or b) extend. A nanny with Montessori training may incorporate that into their style of care, for example by encouraging eye contact when speaking or by using Montessori materials like sandpaper letters. I don't know whether I'd be able to identify a nanny with diversity and inclusion or supervision training, and behaviour management would be a tricky one too as done people seem to naturally have a whole array of strategies and deploy them appropriately. I doubt I could pick out the ones with courses over those who've read extensively but I'm fairly sure I'd be able to identify those engaged in CPD, which doesn't have to be courses as it can be guided reading, over those who haven't. However as there's no requirement for CPD imposed even by professional associations the odd course remains the best way to demonstrate that professional engagement on paper. And yes, I firmly believe CPD improves the quality of care - but again that CPD doesn't have to be courses. It just happens to be, from an employer's perspective, trustworthy evidence. If someone has done the MNT sleep course, for example, I know they've done more than just read a book about sleep training methods. I'm then likely to engage in conversation about what they learnt and that gives them an opportunity to show their theoretical background and understanding, which probably makes me more favourably disposed towards them. Someone who tells me their method of sleep training is CC because that's what they've always done and it's what their first MB told them to do isn't displaying the same attitude. Ditto demand feeding vs routines - I'd expect someone to be able to intelligently engage in the up and downsides of both and probably cite the latest research that demand feeding has been linked with higher IQ.
I accept I'm an anomaly though! I have experience of educational research, working in various EY settings and have studied some of the qualifications myself. I'm probably a nightmare employer but I personally value learning and engagement with theory in conjunction with practice rather than people creating their own personal theories.
That's a mammoth post. Well done for reading.