I think there are several issues here which are clouding things, the two most obvious ones to me are
- You are concerned about your ex's motives
- You are under stress and pressure time wise and can see some benefits to the idea.
Sort out what is important here for you, and what is best for your child overall.
Your child alreay is going to a childminder. Where and who that childcare is with is your responsibility as a parent. If you are going to share a nanny (and lots of people do for various reasons) then you need to think seriously about the consequences and if the arrangement is good and right for your child and for you.
Ignoring the fact they are a family member of the OW... You have to be comfortable with this person. If you were to hire a nanny yourself you would want to know a lot of things about the arrangement first...
a) Have you met them, what qualifications do they have, do they work through an agency?
b) Is this a proper job or simply a fill in arrangement till something better comes along? Who is ensuring this person pays the correct tax for example, is she self employed or employed by the sister?
c) Is there a proper contract? What is the arrangement for holidays etc (do you still have to pay for example of she has 2 weeks off, or if you have a week off).
d) Would you be able to fire this person if you had serious doubts or concerns? Normally you would, but the family connection might make this an issue
e) If things didn't work out would you be able to discuss this with the other person employing the nanny (ie the OW). Will there be any issues of two against one should you want to change the arrangements (ex and OW against you).
Once you decide if this person is someone you would employ to take care of your child(independantly of the fact she is already employed and you'll be joining that) then you have other questions to consider...
a) Where is this nanny to work? There is no reason for example that you can't keep your current childminder and use the nanny two days a week only, or that the nanny splits the time between your house and the OW's house.
Getting home on time is an issue for you. Your ex picks your child up from the childminder 1 day a week, keep that arrangement maybe as it's a day you don't have to rush back. You can then perhaps have the nanny look after her one day at his house and one day at yours and then you'll only be rushing back to the childminder the other two days. Or some other combination.
It can be disorientationg for a child to be swapped between carers, but if that carer is in their life long term it is far easier. They become attached, even to a childminder. This person may then suddenly vanishes from their life. In some ways it would be better to have a family memmber, your ex, mum etc to have your child if possible, because that person will always be a part of their life, a child minder won't. But this arrangement is about a nanny, and although they are a part of the OW's family and may become a part of your child's extended family, they might equally not. This is not about spending time with dad, this is about spending time with a nanny and dad being able to see child for a hour or two extra when he gets home.
There is nothing wrong with your child spending more time with your ex. However the arrangement is for your child to spend more time at your ex's soon to be house with a nanny, not with your ex. Your ex is therefore going to spend how much additional time with your child in reality?
You say you have no residency order in place. Why not voluntarily agree to a SRO (shared residency order) to avoid possible conflict on this point in the future? This doesn't mean the child spends equal time with both parents, but it does mean that the parents have equal rights and responsibilities in all areas. A shared residency order in place makes it less likely that one or other parent will apply to court for a sole residency order in order to resolve the residency issue. If a child has two parents and spends significant time with both then it is better imo for everyone to recognise they have two homes of equal value. This might help dispell some of your ex's concerns over the concerns he raised, plus might help any concerns you have about him having more time with his child in an attempt to gain residency (as others have suggested might be a reason). It also shows that you want him to be an equal and important part in your childs life. Whetehr he's spending more time with your child because of the OW or despite it it must be a good thing for your child. Try and encourange it in any way you can, and hopefully he'll stick with it even if things don't work out between him and the new partner in the long term.
Lots to think about, only you know your ex. Perhaps speak to him about some of your concerns and find ways of resolving them. For example you might agree to sharing a nanny, but the nanny might be better to be someone you both agree on and choose via a proper agency, rather than a relative with no proper contract.
Best of luck
Gilly