As I understand it, women registered at the Jenner can give birth at either Kings or Lewisham, and in fact any hospital they choose, but if you live in Southwark or on the border in the Kings catchment area you can have the Kings midwife come to see you at home. A friend in Dunoon Road did that as she was registered at Forest Hill Road practice. If you are outside that catchment area you are referred back to Lewisham for midwifery services and they dont come out to you. The arrangements however can be a bit odd as are the visiting midwife and health visitor arrangements following reorganisation- our HV was based at Sydenham Green yet our nearest centre was at the Jenner.
Unfortunately I second the negative view about Lewisham both from personal experience the first time around and from what I witnessed. I'll spare you the details however it is not our hospital of choice on any front. Kings on the other hand was great ante natally but as seems to be the case everywhere, overstretched post natally and we witnessed a revolt on the ward I was in as patients partners were increasingly vocal about the lack of attention their pregnant partners were getting and security had to be called in.
Kings still gets my vote however- Lewisham most definitely gets the thumbs down.