It is not exactly a nursery but there is a Casal Estiu - Summer camp for children near the Parc de la Ciutadella in Barcelona in August. It is at the CEIP - infants school.
It runs daily from 9am - 6pm for children ages 3-12 year and costs around 98 euros a week with food. It is one of the public summer camp offerings and has around 100 places in total. There are not that many in August. This is one of the few options.
The contact number is 902 910 050
You probably have to apply from the 10th may as you do with most of the summer camps.
There is info also here on the site about these Barcelona summer camps
http://www.kidsinbarcelona.com/content/view/203/1/