If parents cannot agree on the child arrangements either one can take the matter to court, however first they will have to do a MIAM (mediation) which is a legal requirement before a court will accept an application. See the C100 form for more information on the MIAM and to check if you meet any of the limited exceptions for attending. If mediation takes place, hopefully you will reach an agreement, if it doesn’t, the mediator will sign the C100 and you can apply to court.
The application is made on form C100, and the cost is £215 unless the applicant is on a low income or benefits in which case they may be entitled to fee remission, see form EX160. The court will make a decision based on the best interests of the child. The court cannot, and won't, force a parent who wants the children one day a week to have them more than that however. So if that is your hopeful aim, I don't think you are going to be successful, unless taking him to court shames the father into realising he is not doing his fair share.
Find a mediator here: www.familymediationcouncil.org.uk
C100 form: https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/946762/c100-eng.pdf
EX160 form: https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/952045/ex160-eng.pdf
E160A Notes: https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/958038/ex160a-eng.pdf