Call 211 or 02-730-1000 to have your sim card activated. Make sure you already backed-up your contacts before activating your new nano sim
The activation should only take less than a day (for Postpaid for Prepaid it's 24 hours). I just had my defective sim change a while ago and after I even left the Globe Store it is already active. I suggest that you visit the nearest Globe Store have them validate the new sim with the same number to have it activated immediately.