Southwark Council has made an attempt to get people involved. At each of the 8 Community Councils within Southwark an hour has been dedicated to people with post its giving ideas for increases and decreases in expenditure. Good start. What is letting this down hugely is the lack of any real online way of doing this.
One great online budget consultation is Redbridge Boroufg Council www.redbridge.gov.uk/conversation.
This has been offered free to every local council. Sadly this has not been adopted by Sotuhwark. The best Southwark can do is an email generated form with two cells in it.