I have a personal reason for asking this question. I'm researching my family tree and I have some ancestors who lived in Mile End Old Town, London between 1850 and 1900, but gave birth in Mile End New Town, London which was a nearby but not neighbouring area.
Mile End New Town is in the pinky/purply area at the left, and Mile End Old Town is in grey close to the centre of the map.
I've looked on a modern map of London and I can't see a hospital in the area that used to be Mile End New Town, although the Royal London Hospital (then just the London Hospital) was nearby, possibly in Mile End Old Town.
However, according to this page, the poor did not give birth in hospitals in Victorian times anyway.
It seems women, or at least my ancestors, didn't give birth at home but went to a specific place to have their babies. Where did they go?
Echoing the sentiment in my earlier comment, hospitals were chiefly a venue where the poor would go until the mid-century, and births occurred at home with ad hoc midwives otherwise.
The story behind hand washing before childbirth (an interesting read in its own right) elaborates on the reasons why hospitals attracted the poor:
Maternity institutions were set up all over Europe to address problems of infanticide of illegitimate children. They were set up as gratis institutions and offered to care for the infants, which made them attractive to underprivileged women, including prostitutes. In return for the free services, the women would be subjects for the training of doctors and midwives.
The article further notes that Semmelweis' theories were rather well received in the UK. So I'd suggest that a poor in London in the send half of the 19th century would indeed have given birth at the hospital (or more specifically a maternity institution) if they couldn't afford a midwife, and at home if they could.