Our son was 6 wks old when we got our Mal puppy who was 10 wks at the time.
The puppy and baby do NOT interact. Ever. The puppy isn't a pet for our son, he is a puppy and as such has no ability to be reliable or safe around a child. He sniffs, sees, and is in the same room as our son but that is it.
I would have gone INSANE having a puppy in the house when my son was actually born. Absolutely insane. I ended up having an emergency c-section - totally unexpected and a totally normal pregnancy until right up at the end. I did not so much as touch the dogs for the first two weeks my son was home. My SO took total 100% care of them.
Now, after the first few weeks home, my son was sleeping darn near 24/7 it seemed like so by the time he was 6 wks old and snoozing all day, and decent chunks at night, I was going nuts all day with my SO gone and son asleep. I was used to working A LOT previously so a sleeping baby wasn't at all what I had expected. I spent the first month Dante was home training him constantly. He was housetrained in a matter of weeks, had major obedience training on him very quickly and was already enrolled in classes at 12 wks old.
Now it's a bit more juggling because my son is awake a lot more at 5 mo old now (almost), but my SO works nights and is home with me all day, so it works for us fine.
I would 100% completely and totally discourage you from havinga puppy when you first get home. Those first few weeks home....we were like the walking dead. It was the most exhausting thing we've ever done, and we both worked exhausting schedules prior to our son. Wait a few weeks and reaccess but don't wait too long because once they are out of those first few months they spend a lot more time awake. I got the potty training and basic ob done easily when my son slept so much from about 4 wks old through 3 mo.. JMHO
Forgot to add....we've also raised a ton of puppies between the two of us, and we do a ton of fostering, so training a pup is like second nature. The older dogs set a good example for them too. Not having had a puppy in 10 years, you're going to be in for a shock when you remember how much work they are.