No, not a stupid question at all. It got me to thinking about it.
If no one else is going to have a go, I'll give it a bash!
My records say that a badge would be issued, the early ones being posted out in September, 1916 and were posted out until 1920.
I would have thought that each time a batch of men, with the badge in numerical order on a Roll which was completed, the relevant department would send the badge out to the last known civilian address of the soldier. Of course, the reason for discharge was important. A badge would not be awarded and issued if the man had been discharged through having misdemeanors.
Cause of Discharge is shown under pra 392 KR 1912 XVI or 'No longer fit for military service'.
Discharge treatment details for soldiers with other reasons are laid out in Para 399-410 KR 1912, which includes 'lunatics' and other illnesses.
I have seen the SWB roll pages with only one man included, so I don't think the powers that be waited for so many to pile up. The Rolls were put there and as each page filled up or names stopped coming in, they stamped the badges and posted them off.
As for the men of Southern Ireland, they would have received them, whether they wanted to wear them or not. If they were awarded their medals, a badge would have been forthcoming. They served their King and country, just like all the others who joined up.
The badges came boxed, and with a certificate of issue.
Dick