If you want to move within your current area then you will need to speak to your local authority's housing department - see below.
If you want to move to another area then there are various schemes to help you to do this, but speak to your council's housing department first.
The HomeSwapper website may be able to assist you to find someone in that area who wants to swap with you.
Housing Moves is the Mayor of London's housing mobility scheme to help victims/survivors of domestic abuse who are current social tenants and former rough sleepers moving on from supported accommodation, or other similar accommodation and services, to move outside their existing borough to a different part of London. It is run by the Greater London Authority and the majority of London boroughs and housing associations are participating.
The Seaside and Country Homes scheme is also run by the Mayor of London and provides bungalows and flats for council or housing association tenants over the age of 60 who want to move out of the city to a seaside or country location.
In Kensington and Chelsea the council provides a Mobility Transfer Scheme for people living in ground floor social housing properties, who do not have mobility issues or are very overcrowded but have low points on the Housing Register. These properties would be suitable for tenants who desperately need to be on the ground floor and the council provides incentives to move into other properties.
Exchange Locata (not for Westminster residents) is a direct, online social housing exchange service for tenants and landlords.