Conscription was always the decision of the local government, it wasn't decided by Britain - which (apart from the Royal Navy) didn't really have a legal mechanism to conscript until 1916 anyway, by which time most colonies had enough self-determination to decide for themselves. The British army had always been a volunteer force since the Civil War, and after the American revolution and Indian mutinies, Britain was rather more cautious about making demands from its colonies.

The arrangement varied a lot, depending on time and country. conscription was never implemented in Ireland in either WW1 or 2, either North or South. In Australia, WW1 conscripts couldn't serve overseas, in WW2 the Canadian government had restrictions about where their units could be used - either in the European theatre only for the army, or (after April/may 1945) in the Pacific only as volunteers. New Zealand rules were much the same as Britain, although the Maori weren't conscripted in WW1. Most other countries were volunteer only - most notably India - who in WW2 fielded the largest completely volunteer army in history.