I suspect this is going to be a very ad hoc process, as a PhD isn't a purely taught course and the research is actually the central element to it, supported by the pillar of your PhD supervisor. I think the research training modules element is the least important here at the outset.
The bigger questions I think are around your supervisor(s). Would you be retaining the same supervisor? Do you have a new supervisor in mind? What has your current supervisor said about it, and if they are supportive have you spoken to the new supervisor? Even changing supervisor with the same topic, or changing topic and/or supervisor in the same department I think can be complicated.
There's also funding questions - if you received funding for your current research project, that was on the proviso you would be researching that topic, and may well not be transferable if you change project (even within the same department), much less if you change department. Do you know if your funding will move with you? If not, will you be able to secure funding in the new department?
Really I think the answer for "when a student should start the process" would be "before you start the PhD", or even probably ideally "before you apply to the PhD". For doctoral study changing course like that after starting is somewhat unusual and it's a lot more involved than e.g. switching degree as an undergraduate or masters student. There are a lot of factors to consider even in general, much less anything specific to SOAS itself.