I was also born in Prestbury despite my Mother living in Hulme and attending her antenatal classes at St, Mary's. I understand that St. Mary's has it's origins somewhere near Deansgate before moving to bigger premises at the junction of Oxford St. and Whitford St. (opposite The Palace Theatre and down the side of Whitworth St. towards The Ritz).
St. Mary's was closed during WW2 due to the blitz, so other locations had to be used away from the City Centre, which included Prestbury Collar House. After the war, St. Mary's reopened, but Prestbury was still used due to the post war Baby Boom. When my Mother was due to give birth she went to St. Mary's but was told "sorry, no room at the Inn" and consequently transferred to Prestbury. Apparently, some 14,000 babies were born at Prestbury Collar.
Hope you find it interesting my friend.
Incidentally, there is no shame in having Woodhouse Park on your Birth Certificate :-)