All schools give at least lip service to pastoral care. Many are extremely good at it and all have structures that are expected to support and monitor pupils.
For the most part, the class teacher or form tutor will be the principal person responsible for all aspects of your child’s progress and well-being. Beyond that, the responsibility may fall to the housemaster/mistress or year tutor (or sometimes both), depending on how strong the house system is. There will be staff specially designated as responsible for the school’s PHSE programme, for giving advice on university applications, careers etc and there may be a school chaplain. One member of the senior management team is likely to be responsible for pastoral care across the whole school and, as in all other things, the buck stops with the Head.
Boarding schools will have special staff living with the pupils in the boarding house(s). Schools with a large number of boarders may split them into houses that are the same as the competitive house system, so the housemaster or housemistress lives with the pupils. The boarding staff will have a much broader remit in the care of your child and it is vital that you feel comfortable that the relationship will be a supportive one.