Putting prayers together because of being too busy with breastfeeding

Firstly: 

It is permissible for the breastfeeding mother to recite Qur’aan whilst breastfeeding her infant, and there is nothing wrong with it, even if she reads from the Mushaf, so long as she has wudoo’. Otherwise she should hold it with a barrier such as gloves and the like. See the answer to question no. 10672 and 8950. 

Secondly: 

What is required is to offer the prayers at the proper times as enjoined by Allah. Allah, may He be glorified, says (interpretation of the meaning):

“Verily, As‑Salaah (the prayer) is enjoined on the believers at fixed hours”

[al-Nisa’ 4:103]

i.e., they are obligatory and must be offered at specific times. 

It is not permissible to put two prayers together except for a reason such as travelling, rain or sickness. 

Ibn Abi Shaybah (2/346) narrated that Abu Moosa al-Ash‘ari and ‘Umar ibn al-Khattaab (may Allah be pleased with them both) said: “Putting two prayers together without an excuse is a major sin.” 

Some of the scholars said that one of the excuses that make it permissible to put two prayers together is: “… for the breastfeeding woman because of the difficulty caused by there being a great deal of najaasah (impurities)” — i.e., because it is difficult for her to purify herself for every prayer. Abu’l-Ma‘aali said: She is like one who is sick.

End quote from Kashshaaf al-Qinaa‘, 2/5. 

i.e., if there is some impurity (najaasah) that gets onto her from her infant, and it is too difficult for her to purify herself or change or wash her clothes, then it is permissible for her to put prayers together.  

But if she does not get any najaasah on her, especially nowadays with the availability of diapers (nappies) that keep the najaasah from spreading, it does not seem that there are any excuses. Doing wudoo’ and praying takes only ten minutes or a little more, so it does not make sense that a woman could be distracted with her infant to such an extent that she cannot leave him for this short period of time. She should take advantage of his naptime or times when he is calm to offer her prayer in the best and most complete manner. 

Prayer is the greatest act of worship and the most important pillar of Islam after the twin declaration of faith (Shahaadatayn). The Muslim is enjoined to pray regularly and to pay attention to the matter of prayer and give it its proper measure, because it is the connection between him and his Lord; if he neglects it there is no greater loss than that. 

So our advice to you is to pay attention to your prayer and offer prayers regularly on time; remember that this is one of the means of barakah (blessing), protection and divine support, as Allah, may He be exalted, says (interpretation of the meaning): 

“Whoever works righteousness — whether male or female — while he (or she) is a true believer (of Islamic Monotheism) verily, to him We will give a good life (in this world with respect, contentment and lawful provision), and We shall pay them certainly a reward in proportion to the best of what they used to do (i.e. Paradise in the Hereafter)”

[al-Nahl 16:97] 

“And whosoever fears Allaah and keeps his duty to Him, He will make a way for him to get out (from every difficulty).

3. And He will provide him from (sources) he never could imagine. And whosoever puts his trust in Allaah, then He will suffice him. Verily, Allaah will accomplish his purpose. Indeed Allaah has set a measure for all things”

[al-Talaaq 65:2-3]. 

And Allah knows best.