These tend to be national and often multi-species. Most have local centres and some of these are on Twitter.
Dogs’ Trust http://www.dogstrust.org.uk/
Cats’ Protection http://www.cats.org.uk/ Cats’ Protection also run the Cat Rescue Awards http://www.cats.org.uk/get-involved/rescuecatawards/
Animal Accident Rescue Unit http://www.animalaccident.org.uk/ This is a Nottinghamshire-based charity dedicated to rescuing animals injured or in need of medical attention. Twitter http://twitter.com/AARU_Nottingham/
Animal Welfare Network Hub http://www.civicdirect.com/topic-hub/animal-welfare-network-hub Twitter http://twitter.com/animalwelfarecd/