It appears you senior managers and company have really dropped the ball on this. There should have been clear guidance issued to managers and staff.
Grievances are raised. Not always reasonably. Having a grievance raised against you in the past isn't an excuse to not manage, which is what your senior team seem to be doing. As a manager when one is raised against you, its either unreasonable and the employee is the problem. Or its a reasonable grievance and you learn from it and move on.
I am shocked that this person has been a poor performer and been allowed to get to 2 years employment without her performance being handled.
Its not that childcare can't be mentioned or has some protected status. The reason I wouldn't mention it, is because there doesn't seem to be any evidence that this is the reason. Leading with that, will just enable her to keep bringing back to that and you have no proof.
Get HR involved to support you in starting a Pip. You can't just go into a pip. There's alot of prep work before hand.
How aware is she of her poor performance? Are you having conversations with her about this? Did her previous manager tackle her poor performance?
Does she have clear targets? So you say she doesn't answer phone calls and calls back? What is it you want her to do? What's the target?
The urgent email? What turnaround time is expected? You can't just say 'respond to urgent emails quicker' or 'you don't respond to urgent emails quick enough'. Do you or your company actually have targets or response times? Do the rest of the team have the same targets? Are they achievable and do the rest of team hit these targets?
Are you sure you are aware of everything available to you to be able to track wether people are actively working? After over 18 months of home working, I would be surprised if nothing is available. Our system shows us at a glance if people are active. But then there's layers, we can see which system they are in. Down another layer which part of the system. Eventually, if needed, IT can retrieve actual key strokes made.
If you are going to remain WFH the company really needs to be able to have something. Micro managing is a pain the behind and I rarely use anything other than the top layer to see everyone us active. But sometimes you do need more available