I totally agree with Lee-Bee.

Contract should have firm start date and their start date isn't subject to what suits you but when they need care. Negotiating these detail should happen before contract and money exchange.

It clear e-mail isn't getting you answer in time frame you prefer but they likely away. If I wasn't starting work until September, I would be making most of family time before and not checking in with things that a month away just because someone wishes I was.

Phone the house and try and speak to her.

If she needs care from 11th, it seems really obstructive to offer next day! Why not offer earlier date since not having two on same day is totally about your preference and convenience. I'd be fuming if I paid back in March to secure care, was asked when I wanted to start and then was told I would have to wait until next day or worse, next month!

Change your contracts so firm start date is on them and all this can be avoided in future.

It seems like poor planning and making things over complicated.