I had my third baby when I was ~18 months into my apprenticeship. I brought her with me to prenatal/postpartum visits for ~10-12 months, but not to births. I think it would have been fine taking her to births until she was 3-5 months old, but after that it would have been hard. I did take her to one birth that I did for a repeat doula client (having her first home birth, with a midwife other than my preceptor, and I was just there as support/photography--the midwife had her own assistant there). It was really hard. The client had become a friend, and was totally okay with the fact that I was constantly in-and-out checking on the baby, or that she was babbling quietly on my back in a carrier for part of the birth, but I wouldn't make that assumption about most clients.
I'm 37 weeks pregnant now, and still apprenticing with the same midwife that I was when I gave birth 2 years ago. We'll have a similar arrangement--I'll bring the baby to visits, but not births (with the exception of one repeat client who is due ~7-8 weeks after I am who asked me to come and bring the baby--I won't be officially "back" at that point). I'll start attending births in January/February.
As far as prenatals/postpartums, I'm sure it has a lot to do with your baby's personality/temperament. It was really easy for me to have my baby with me before she was mobile--she was always happy to nurse/be held, or sometimes to just lay on a blanket on the floor. When she started crawling, it was still all right--my preceptor's office is pretty child-friendly--but she did start "getting into stuff" at that point. Still, it wasn't much of a problem.
For me, she became considerably more difficult to take along when she started walking. If we had been at the office more, it wouldn't have been too bad, but we do home visits from 36 weeks on, and obviously not everyone's house was toddler-ready. It didn't seem to bother our clients too much, but I often felt like there was no point in me being there, since I gave most of my attention to chasing her around and trying to get her to stay out of cat food, or whatever.
I think you really have to just play it by ear, so to speak. My ideal set-up would be to take the baby to prenatals for the first year (maybe a tad longer if you were primarily in an office, rather than in homes) and to births for the first 4-or-so months. I would not want to do another birth with a 6+ month baby, especially if I had any "real" responsibility in terms of the birth.