It would of course be wrong and a breach of duty to knowingly accept
instructions from another party in the case having begun acting for the
first party, and I suppose that might happen due to a breakdown of
record-keeping in the firm. Quite often the concept of "chinese walls"
would enable several lawyers in the same firm to act.
But if the interests of the client conflict with the interests of the
lawyer, eg if the lawyer is being asked to put forward a case that would
put him in breach of professional standards, then I'd call that a
conflict of interests. The lawyer in that situation would not want to
undermine his client's case by explaining to the court the precise
reason for recusing himself.
It should be quite easy to find a new law firm to represent the son, and
that law firm would make whatever application is necessary for an
adjournment, if indeed an adjournment is necessary. The case might
perhaps be ready for trial and most lawyers are capable of taking over
conduct of a well-prepared case. Sometimes barristers only meet their
client on the morning of the trial.