The (Muggle) UK Child Care system is run by local councils. The relevant department used to be Social Services, but fashions change and these days they are usually called Children and Young Peoples Services. The short answer is no, it would not
Abandoning a baby? Leave it outside a hospital, preferably one with a maternity unit, and leave it somewhere where it will be found. After that the NHS system will take over, and the baby will make the local news. The child will be found a foster home, but the lack of paperwork (no parents to give consent) will, I think, slow down the adoption process. I’m no expert.
Goldensnidget covers education in detail, except:
The system is local and every council can do things slightly differently. In the council areas immediately to the south of me many (not all) Primary schools are split into two schools
Infant School (up to Year 2)
Junior School (Years 3-6)
In other places (like where I live) they ignore Primary/Secondary use a First/Middle/High school system
First School (up to Year 4)
Middle School (Years 5-8)
High School (Years 9-13)
These alternatives are quite common in rural parts of the country, as they keep the younger kids in smaller schools closer to home.
To the north of me lies Scotland.
None of the above applies to Scotland (I’m sure a Scot will correct me if I’m wrong). As I live on the border, my understanding is this.
The age on starting school is different, the naming system is different (most “secondary” schools are “high” schools), and the exams are different.
The exams are Standard Grade, Intermediate Grade, Advanced Grade and Higher Advanced Grade (I think).
I suspect that all I’ve done is confuse you, but it’s as well to remember that a lot of answers in “Being British” should carry the warning: “Applies to England and Wales, Scotland may be different (and Wales may be different, too).”