In the UK the terms 'custody' and 'access' were replaced with the 'residence' and 'contact' some years ago.
The first thing to establish is whether the father was named on the birth certificate and has Parental Responsibility. If so he may apply to court for any order relating to PR including residence or contact. If not, a PR agreement can be made or it might be ordered by the court. Grandparents require the permission from the court to apply for any order.
Most separating parents have PR or are eligible to apply for it, are seen as equals in the eyes of the law and come to arrangements about where the child should live and how much contact between themselves. Experience shows that although there might be some posturing (unfortunately some solicitors encourage this with an adversary approach) in the short term couples are more satisfied with arrangements they made themselves and are more likely to adhere to them.
When courts become involved they decide arrangements for children based on a welfare checklist. Generally it is accepted that it is in the child's interest to maintain their sense of security and existing bonds. Therefore if one parent works full time and the other doesn't work or works part time the reality is that the children has a parent with the majority of care and a contact parent. When both parents work full time and child care is shared it is a shared residence situation. Shared residence doesn't have to be 50:50, it can be divided in different proportions.
When it comes to contact parents have no rights, it is the child's right to a relationship with the parent and their extended family. With very young children contact tends to be for frequent short periods. Over time this develops in to days, overnights, full weekends and fortnight holidays. By the time children go to school sharing the quality time at weekends and during the school holidays is fairly common.
My suggestion would be to sit down and try to draw up a parenting plan possibly with the help of a family mediator. There are lots of examples on the internet and CAFCASS publications available to download from their website is a good starting point. A parenting plan cannot be made legally binding but some solicitors will draw up a deed of agreement which can be referred to court should there be disputes in the future.