I don't know the answer - but if you look up some of the agencies that employ social workers such as Reliance Care or telephone to the employers advertising the jobs - then they would be able to tell you if they are prepared to take someone one with a work permit.
Many places don't, however, if they isn't anyone suitable for the post - then they will consider taking on someone who needs a work permit - this procedure applies for the NHS - not sure about social work though.