I hope it all works out for you. I did this (ish), tried to finish all my work before the little one arrived, but the uni messed up so I ended up writing a dissertation with a newborn. It was reeeally hard! However, it will depend on the baby, some are easier than others, some sleep better, some fuss more, and it will depend whether you use formula or breast feed or a combination and whether you have childcare lined up.
I breast fed, couldn't express, and I had a very hungry baby! He was not a good sleeper, he was very demanding. I asked the uni for an extension and after a lot of hassle eventually got one.
Mine was a distance learning course. I could leave the little one with parents for a couple of hours at a time until he would need feeding again. But they weren't that keen on taking him, and I wouldn't have felt comfortable doing this when he was really little, I started when he was as a couple of months old. Most childcare providers don't take them under six weeks. If you get a decent sling you might find that for the first few weeks your little one is happy just spending a lot of the day sleeping on you which could help.
I would say it would be easier with a one year old as it's easier to put them in childcare. But then I know some people who were able to work from home whilst caring for a baby, I couldn't have done that with mine as he was too demanding.
I think it all just depends on the baby, you, how you feed and how much support you have.
Good luck and I hope it goes well!