If one day the child goes around to the father's house, perhaps in a
state of upset, and says, "That's it, I'm staying," what is the legal
position of the father?
An eleven year old child really has no say in the matter. A judge in a custody case may consider what the 11 year old has to say but is unlikely to give it much weight. (In contrast, a judge is likely to give a lot of weight to the views of a child who has a job, who is doing O.K. in school, and is a year or two from becoming a full fledged adult.)
This is up to his parents to resolve absent circumstances not present here (e.g. the child has made bona fide allegations of child abuse, or the parents are both incarcerated). And, if the parents can't resolve the dispute, it is up to a court in a case where parenting time is at issue (probably either a legal separation or a divorce in this case).
For example, while (as noted below) the police will not generally drag a child kicking and screaming to the other parent without a court order, if the child is at his father's house, the father can absolutely drag the child kicking and screaming to the child's mother's house, no matter how much the child doesn't like it.
Moreover, while the father will not be violating any law, if he does not do that, a child custody judge is likely to look dimly upon a parent who intentionally withholds visitation from another parent without good cause, when the court considers what kind of child custody arrangements to put in place. And, the court has extremely great discretion in these matters.
The judge is also likely to be pretty unhappy with both of the parents for failing to be capable of communicating or cooperating over child rearing related issues, because they are apparently so focused on not getting along with each other over their issues with each other.
In particular, if the mother calls the police claiming "child
abduction," what will be the police position? Will they come to the
door of the father? Will they drag the child kicking and screaming
back to mother's house?
It is not generally illegal for a married parent in a situation where there is no child custody order in place to have that parent's child with them over the objection of the other married parent.
Absent a court order to the contrary, the police will probably call this a "civil matter" and will be unlikely to take any action until there is court guidance, absent exigent circumstances like evidence of recent not previously adjudicated claims of child abuse or neglect.
The way to resolve this would be with a court filing of some sort seeking to resolve the parenting time issue, if necessary, on an emergency basis.
Whether or not the parents want to end the marriage, this situation is long overdue for court intervention. I've had couples who "pull the trigger" too soon, but this would not be a case of that type. Either the father or the mother needs to find a solicitor (if at all possible to afford that) and get the court system involved.
If there is a breakdown in communications over matters related to the child, and the child is not cooperating in the face of an ambiguous parenting situation involving two parents who can't communicate about their child, the situation is out of control. Prompt court action could prevent a more negative outcome in the future by creating stability and structure in the situation.