We're all busy, no matter what our situation but for what it's worth I'm a SAHM with 3 DDs aged nearly 5, 3.5 and 10 months.
If I were to knuckle down, be brave and sort myself out I could start a part time Masters in September. By then DD2 would be in reception. Attendance at university one day a week, DD3 in childcare for that day. DH works away a lot of the time. Reliable sleep is still some sort of distant dream although could have improved by September.
I have some sort of reassurance from my not entirely reliable dad that he could come over for a day maybe 3 weeks out of 4 to give me some time to study during the day.
Tell me how horrendous it could get! Or tell me that you managed it while also working full time with no help to persuade me that I can do it.