Fairfax had endorsed Monk. He had of course never supported the execution of Charles I, despite being named by the Army Council to be President of the Court. As his wife reportedly said at the trial when his name was called "He has more wit than to be here".
Lambert tried to seize power. He thought that defeating Booth at Winnington Bridge made him heir to Oliver's mantle. However, the taxpayers "went on strike" so that his men found themselves without pay, while the Navy repudiated him and expressed their feelings by blockading the Thames. When he went north to confront Monk, his unpaid troops deserted. Somehow I get the feeling that he wasn't really up to the job.