One parent will be put in an adjacent seat (might be across the aisle or behind/in front), and two parent/child pairs may be a long way from each other (e.g. 20B/C and 38E/39E or whatever, that kind of things) if you do not pay for advance seat assignment*.
But BA will do what they need to meet their legal obligations, which means at least each child will be seated in an adjacent set to one of their parent, but BA are not obliged to do more than that.
If you really want to seat the whole family together, then you'd need to pay for it in order to assure being seated together. However, it might happen that BA will seat all of you together anyway without fees. Of course, it's entirely possible that kind-hearted passengers** who notice you being separated may offer to swap on board so that you can all be seated together as well.
* In the absence of an appropriate OW status.
** There is a degree of self-interest in this as well, in that a family seated together tend to be quieter for those around them, as they won't be shouting to each other, moving around the cabin to spend time together, or kids missing one of the parents seated away from them etc, so don't feel too guilty if someone offers to swap
