It would be great if you could share the exact message that appears when you try to install the mod, as it seems that the system may be requesting approval from a parent account before allowing the child account to sign in.
If you changed the email address associated with the child’s Microsoft account, you should still be able to use the new email address to sign in to the same Microsoft account on Microsoft websites and services.
If, instead, you mean that the email address was also being used to sign in to a third-party service, you may still be able to use that same address there, depending on how that service handles sign-in. However, if the third-party service requires you to verify access to that email address and the address was permanently removed from the Microsoft account, this could become a problem.
Please share the exact message or a screenshot of what appears when you try to install the mod so I can better understand which account is requesting approval.