If you move into a house where a cat or dog has spread dander it can take up to two years to clear the allergens. They stick to walls and are in the carpets and soft furnishings. If you suffer any form of allergy then you should consider removing the pet from the home. If this cannot be done then confine the dog or cat to one room. The kitchen is best because it has no soft furnishings.
Some people are so allergic that they have only to walk past a cat to set off an asthma attack.
The immune system has been primed and you might start to see wheezing when the child is about two. First - a visit to an allergist, to be tested and see how clinically severe the allergy is. Third - install air filtration in the home. Fourth - realise the practical limitations of interaction with the dog. I adore my dogs but as an asthmatic I don't snuggle them much, I don't bury my face into them, like many owners I know.