There are some services with unlimited demand, like healthcare. My guess is that the demand will switch to other services as well, like the nannies you mention

There are some services with unlimited demand, like healthcare. My guess is that the demand will switch to other services as well, like the nannies you mentioned and better education and other wealthy-person services: personal trainers, nutrition, leisure, party/wedding planners, clothing advisors, gourmet cooking. There will also be a lot of services needed for the elderly. Of course, competition will be cut-throat, but those jobs that cannot be made more efficient or paralellized (those with a high amount of in-person involvement) will sustain the new middle class.