Traditionally, a barrister would receive a brief from solicitor, so would not enter into a contract with their client and so could not sue if their fees were not paid. Similarly, the clients could not sue barristers for breach of contract.
However, under the directed access rule, clients can instruct barristers themselves. In such a case, the client could sue a barrister for breach of contract and the barrister could sue the client if fees are not paid.