Upvote:1
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.