Hello Leon!
For muggleborns it depends on whether they are in school, at an orphanage, or have parents that already know about their child having magical tendencies. During a blood census, everyone was required by law to 'register' at assigned Ministry locations. There was even random blood testing that occurred over the years to check on the validity of everyone's registration.
Schools and orphanages likely would have had specific registration days/testing days throughout the year. If Leon is fairly young, then he might have been at an orphanage all of this time, in which case the registration, testing and badging would have been done for him by his caretakers.
This law was put into place in 1937, and with registration a badge would have been provided.
HOWEVER, and this is exciting for so many, with our new Minister of Magic Pryce Hir, the wearing of Blood badges were removed from law! Leon would still have to register during census, but he no longer has to wear the red M.The law hasn't yet been changed
so Leon and all citizens will still be wearing badges. *blush*
