I was referring to both and I have dealt with experts within this field.
CLAPA : Cleft Lip & Palate : FAQs: What treatment is available?

A cleft lip is usually surgically repaired by the time a baby is 2-3 months old. It requires a general anaesthetic and takes roughly one and a half hours. The surgeon re-arranges the skin and muscles of the lip so no skin grafting from other parts of the body is needed.

Most babies recover very quickly and will not experience much pain after this operation. Medication is given for any discomfort. The hospital stay for this operation is between 3 - 5 days. A parent is usually welcome to stay in hospital with the child.

We were under a Surgeon at Wordsley hospital in the Midlands,well known for plastics work,this hospital has now closed and my son is currently under Birmingham Children's Hospital.
Again Birmingham are saying 3 months,for the cleft lip to be closed.

West Midlands Regional Centre For Cleft Lip & Palate | Surgery

An operation is performed only when the baby is healthy and not suffering from a cold. It is also important that your baby does not have any vaccinations in the 2 weeks before and after the operation. By 3 months of age, most babies with a cleft lip will be big and strong enough to have it repaired.

Feeding after the lip operation is not usually a problem. The nurse or the speech and language therapist on the cleft team will be able to answer any questions about feeding.

And again this site has a timetable of when things happen,which is not immediatly as you can clearly see,
Cleft lip and palate Treatment - Health encyclopaedia - NHS Direct
In England, all children born with a cleft will be referred to a specialist cleft clinic.

Specialist cleft clinics are dedicated medical centres that are designed to provide life-long support to children and parents.

Working at the cleft clinic will be a cleft treatment team. The team is made up of the healthcare professionals listed below.

A specialist cleft nurse - who can provide initial advice about feeding, and will act as your first point of contact between yourself and the rest of the team.
A surgeon,
A speech and language therapist.
An orthodontist - a dentist who specialises in the reconstruction of the teeth.
An audiologist - a health professional who specialises in assessing and treating hearing conditions.
A psychologist.
A paediatrician - a health professional who specialises in treating children.
Care plan timetableMost children with clefts will receive the same type of care plan, but it will be tailored to meet their individual needs. A typical care plan timetable is described below.

Birth to six weeks - hearing test and feeding assessment.
Three months - surgery to repair the lip.Six months - surgery to repair the palate.
Nine months - post-operation hearing test.
Eighteen months - speech and language assessment,
Three years - a further assessment of speech and language

So I would hazard a guess and say unless you are in a position to pay private,corrective surgery is not done immediatly but around 3 months old.
Do you have a link you can post rather than me having to go looking for it ?