Fathers of children born outside of marriage and not named on the birth certificate have no legal right to their child and would have to apply to the Courts. The mother always has parental responsibility.
When you make a Will you have the option to name a Guardian(s). They can be appointed to act upon your death or after both you and your partner have passed away. If the Father does not have automatic parental responsibilities but you would like him to then you can nominate him as your first choice Guardian. You may even wish to appoint an older child (over 18 years) to act as guardian to a younger sibling.
Through writing your Will you are also able to make provisions for those you nominate as Guardians to ensure that they are able to carry out your wishes.
Ultimately you have the choice to choose someone you feel would be suitable and not leaving the decision to a Judge, Social Services or a distant relative!