The Aylesham Centre - Public Consultation Hub. The Aylesham Centre Peckham, in the London Borough of Southwark. Berkeley Homes is developing the vision for the future of The Aylesham Shopping Centre and would like to hear your thoughts on the needs and priorities of the area to help us shape the early ideas for the site. Visit our website for the latest updates on the Peckham shopping centre and Aylesham Centre redevelopment.