No, it's nothing to do with dairy farming. A "milkround" is a process that the larger firms go through each autumn when they are recruiting for training positions. Usually the milkround process takes place in colleges and universities around the country, focusing on the graduate classes of degrees in accounting, finance, and other relevant areas.
Traditionally these firms screen applicants, invite them for interviews, and make offers before the end of the year. It is intended that the new trainees go into a Training Contract with their training firm the following year.
Other firms and businesses that do not participate in the milkround advertise later in the yearly cycle, or they may take on students on an ad hoc basis over the course of a year. Similarly large companies with graduate programmes will hire students (often into the Flexible Route) at various times in the year, usually before summer.
If you haven't had the chance to participate in the milkround process, or you would prefer to take some Chartered exams before beginning your training, you can enrol into the Flexible Route. For more information on this route, please visit the entry routes page.