Nanny housekeeper

You might want to give a general location if you want live out.

Would the person be expected to clean plus look after the baby full time?

Have you tried any of the agencies?

Tbh a good nanny with lots of baby experience is unlikely to want to be a housekeeper

A good HK who can clean may not be good /experienced with babies

HK/nannies generally work better when older children as babies take up more time

You may need a cleaner and a nanny

I think if you want a cleaner who does some babysitting you'll struggle if you employ a nanny. Nannies usually have a very defined role, hope you find what you're looking for

Nanny housekeepers are very common in London. A quick look at the main agency near me has several vacancies for them, and several candidates listing willingness to take live in or live out Nanny Housekeep roles. It seems more common for live-in though.
Anecdotally, non-English nannies seem to the ones who want the roles, so if you've got your heart set on a Norland, it will probably be harder.

You're probably best off going via an agency. There is no shortage of them in your area.

Little Ones is the agency to try, or Eden. They are so highly sought after in prime London they do earn very well.