Ho messo solo le righe principali per non postare la funzione intera.
Prova così:

codice:
procedure TMainFormServer.IdTCPServerExecute(AContext: TIdContext);
            ...
            TFSFileIn:= TFileStream.Create(Nick+ '.'+ FileName, fmCreate);
            try
              try
                ...
                AContext.Connection.IOHandler.ReadStream(TFSFileIn);

codice:
procedure TChat.TMISendFileClick(Sender: TObject);
var
  TFLFileOut: TFileStream;
  ...
begin
      ...
      TFLFileOut:= TFileStream.Create(TODFile.FileName, fmOpenRead);
      try
        try
          ...
          IdTCPClient.IOHandler.Write(TFLFileOut, 0, true);