It's just as easy to envision the House where Redcaps outnumber the magical practitioners more than 10 to 1, that the House can have a well developed system for fostering the scion of their magical members. It may very well be that some of the magical Mercere want to keep their children close, but some might not. Children of magi who aren't themselves Gifted might be quite capable of being Redcaps in their own right, or even doing other work related to the House's function, without actually being called a Redcap.
But I'd say that, just as any adult can "get used to" a wizard, and learn to trust them, a newborn would bond with its mother almost immediately and without question. There might be rare exceptions (and some ill effects - LOTS of crying, I'd think?), but the love would be almost an "essential nature" sort of thing.
Almost. Always room for exceptions for a good story. imo.