In England & Wales the law is framed so it's the child's right to contact with the parent, not the other way round. The presumption is that if the children live with one parent, seeing and knowing the other is in their best interest unless there are exceptional reasons for no contact. Children need a clear understanding of their parental heritage or what has happened in their lives to develop their self esteem, sense of self and attachments - failure to do so ultimately increases the likelihood of dysfunctional relationships, teenage pregnancy, emotional ill-health, etc.
However, one of the factors the court needs to consider when making an ordering in children cases is the wishes and feelings of children, with their views been taken into account depending on their level of understanding and maturity. Most children's views will carry weight by the time they are 12 years old and gradually as they grow older their views will carry more weight. Contact orders normally end when a child reaches 16.
Also after a gap it is common to start off with the odd hour or so contact, possibly supervised, and gradually build it up over sometime. Thus the children can then get used to the idea of contact again and the non resident would need to show commitment at each stage before progressing to the next.