The underdeveloped nature of a baby's throat, with the epiglottis being much higher up in the throat means that squirting anything up an infant's nose - any excess beyond that which coats the nasal passages are likely to be inhaled to the baby's lungs. See: Choanal atresia.
This is of course dangerous, and should not be attempted without the supervision of trained (orthodox) medical professionals.
As to the efficacy, if the baby caught the cold from the mother, by the time the baby is showing symptoms, the mother's milk may well contain appropriate antibodies. The question you would then need to ask is will these antibodies affect viral propagation in the nose/throat/ducts to eyes and ears and the variety of sinus cavities which the virus can affect. I know of no research which shows that this is possible or even desirable. See: Alternative medicine for colds.
Systemic treatment by introducing the antibodies into the digestive tract (the "yummy-mummy" way) is the natural way which has evolved to protect the baby. One might imagine that direct treatment of the nasal-mucosa - if it were to have any strong survival bias in evolutionary terms, babie's noses would be adapted to clamp onto the nipple and provide a sucking reflex - no such adaptation exists. This is another reason to question it's validity.
If the elder sibling is producing milk, then it may contain the specific antibodies necessary to buffer the baby/infant's immune system. If the mother doesn't catch the cold, this would indicate (probably) that those antibodies are already present and yummy mummy would help protect the younger sibling. Timing is important, if the youngster catches the cold before the mother shows symptoms, it's too late for yummy mummy as a first line of defence. See: Cold Prevention.