Twisha - I think you are getting a lot of negative comments on here that aren't really fair to you. I see what you are saying and I understand your concerns. Maybe this is because I am a mother of grownup sons and a GM of 5 children. I know that it can seem like you are over involved with your son and his partner's relationship and they are both adults and need to sort it out themselves, but it is difficult not to be concrned about our kids, whatever their age.
It sounds to me like your son's partner is a troubled young woman who has had a difficult past, probably a difficult childhood and she clearly has had troubles with her previous r/ships. She may not have a mental illness as such but it may be that her life experiences have left her short of emotional maturity and a rather unstable disposition. Your son sounds like he is quite passive and a bit afraid of her and she exploits this by controlling him. Thing is though he is allowing this situation to happen and he needs to change his behaviour, rather than trying to change hers.
There are lots of posts about women living in these kind of r.ships with controlling men, but this way around is more unusual. There are clearly no worries about the children as your son cares for them.
Just on the legal issues, there is no such thing as custody now. If parents separate, and they cannot agree who should care for the children, then the court has to decide. The Order that is made to determine with whom the child lives is called a Residence Order and if the parents can't agree then the court have to make the decision based on reports made by social workers who will undertake assessments. The court then makes a decision and makes a REsidence Order to one parent or the other and in almost all cases, awards contact to the other parent.
From what you say, it sounds like your son's partner might not object too much if he did want to apply for a Residence Order, but of course there is no way of knowing.
Best advice I can give is to be there in the background to offer whatever support you can to them both. Maybe you have to back off a little if only to save yourself getting too worried. Or maybe advise your son that unless he stands up to his partner, nothing will change and she will continue to control him.