In many cultures, the role Alfred fills is more similar to that of a mother. The father, until recently, wasn't supposed to do much emotional bonding with his children, he was the one who was supposed to teach them. In that sense, Bruce is an almost stereotypical father, and Alfred an almost stereotypical Team Mom.
hence why he adopted Jason right away and Tim as soon as soon as he’d been orphaned.
This is just an interpretation, but I'd be willing to bet anything that Bruce wouldn't have adopted Tim if he hadn't adopted Dick before or if Dick had died after Bruce adopted him. It took him a long time to get over the fact that Jason had died; when he adopted Dick and realized it didn't immediately sign the death warrant of another loved one, it became safe to express his love that way. Safe to ask Tim when the option became avalaible, and safe for Cass later on.