Never agree to give up your half of a home without taking advice first. If you become homeless and you have voluntarily given up your home the council may decide you are intentionally homeless. There will then be no duty to find you a new home.
If you are the sole owner you have an absolute right to the property.
If you are married to the sole owner you have a right of occupation of the matrimonial home but any financial interest in the property would have to be decided through a solicitor or the courts.
If you are living with the sole owner you may have a few rights. If the relationship is long term and you can demonstrate that you contributed to the mortgage or household expenses you may be able to show a financial right to the property. You should consult a solicitor about this.
You may be able to get a temporary right to occupy through an Occupation Order.
If you are a joint owner you have an absolute right to live in the property unless a court order has removed that right.
To transfer ownership to one of the joint owners is very complicated – much more so than for a tenancy – and you should always consult a solicitor in this case.
You cannot use the Family Law Act. You can use The Matrimonial Causes Act and sometimes the Children Act.
A court would weigh up whether it is reasonable to transfer ownership and whether you would be able to afford to remain in the property.