Since the end of the second Boer war all British empire and commonwealth forces utilised standardised British equipment, organisation and training.
In terms of training, this was governed by various articles and official expressions of doctrine- field service regulations 1937, 1939 (FSR), Military training pamphlets, Infantry training parts I-VIII (1944) and various annual training memorandum (ATMs). In terms of how effective this training was, the forces deployed were trained at the same level of the British army. More information on training would be found in unit and formation war diaries for primary sources, or in army specific books-e.g. place's infantry training in the British army. As far as I know, there is no book that carries out an empire wide case study