Nanny-Housekeeper Conflict

We've had our current nanny for about a year and a half, and our housekeeper (3x/week) for about 2.5 years at this point. From the get-go our nanny was not the biggest fan of the housekeeper's work, with regular complaints about XYZ that she wasn't cleaning properly. Our housekeeper, for her part, has definitely engaged in a silent war where she will not touch certain cleaning tasks that she considers our nanny's responsibility.

For context, both of these women are older; in a previous life our nanny was a SAHM who employed (possibly full-time) cleaning help, and our housekeeper used to nanny full-time.

It's becoming clear that we're going to have to let our housekeeper go soon, as she has slipped from not-amazing performance to truly substandard. I was hoping that we would be able to postpone a transition until after my maternity leave starts, but it's looking like my husband is not willing to wait that long. This means that the only person who is going to be present during trial days is my nanny, with my husband or I available to say hi/bye at the beginning or end of a shift.

My nanny is pushing both for a replacement and wanting to "train" the replacement; I'm a little bit leery of what the consequences of allowing her to take free reign in this way would be.

For those of you who have experienced workplace conflicts with other household staff, or been involved in the hiring process of new household staff: anything I should be aware of during this process? I wouldn't necessarily mind if my nanny had more of a supervisory position to a new housekeeper, as opposed to the current coworkers relationship, as she is definitely more aware of all the cleaning tasks than I am, but what am I missing here?