His wife is breastfeeding; should he delay Hajj until next year?

If you have already done the obligatory Hajj, then you have
the choice with regard to a voluntary Hajj between doing it this year or
delaying it until next year, because of what you mentioned about being busy
with your family or wanting to stay with them and so on. In that case you
can choose not to do it at all, because it is voluntary and not obligatory. 

But if the question is about the obligatory Hajj, then it
depends. There are two scholarly views, one of which is when one becomes
able to afford it, Hajj becomes obligatory and must be performed in the same
year; the other view says that it may be delayed. However, the more correct
view is that it becomes obligatory with immediate effect and must be
performed in the same year. The one who can afford provisions and a means of
transportation is obliged to do Hajj, and it is not permissible for him to
delay it. Please see question no.
41702 

Based on that, if your leaving your wife will not cause her
or the child any harm, then you must do Hajj this year and you do not have
the option of delaying it in the hope that your family could go with you.
But if your wife has enough money to do Hajj, or you can pay the expenses of
Hajj for her, and it will not harm her or her baby if she goes to Hajj, or
she can leave the baby with someone who can look after him, then she is
obliged to go for Hajj. Otherwise, it is permissible for her to delay it. 

The point is that Hajj may be obligatory for the husband but
not for the wife, depending on whether the conditions are met or there are
any impediments. So the husband should not delay his obligatory Hajj because
of wanting to take his wife with him for Hajj. 

Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) was asked
the following question: My wife has not yet done the obligatory Hajj, and we
have a baby who is four months old and he is being breastfed by his mother.
Should she do Hajj or stay with her baby? And is it better, if she goes for
Hajj, to take pills in order to stop menses, or not? Please advise us, may
Allah bless you. 

He replied: If the baby will not be affected or harmed by her
leaving him, in that he can be breastfed by someone other than his mother,
and there is someone who can look after him properly, then there is nothing
wrong with her doing Hajj, especially if it is the obligatory Hajj. But if
there is any fear for the child, then it is not permissible for her to do
Hajj even if it is the obligatory Hajj, because the breastfeeding mother is
permitted to not observe the obligatory fast if she fears for her child. So
how can it not be permissible for her to postpone Hajj if she fears for her
child? If she fears for her child, then what she must do is stay, then when
he is older she can do Hajj next year. There is no blame on her if she stays
home and does not do Hajj, because in this case she is not obliged to do
Hajj immediately. 

With regard to using pills during Hajj or ‘Umrah, there is
nothing wrong with that, because this is a need. But it is essential to
consult a doctor, because the pills may be harmful and may cause her harm.

End quote from al-Liqa’ ash-Shahri, 10/25 

And Allah knows best.