You can't 'wake up' your Tama, unless you set the clock to the time it should wake up. Different characters have different sleeping times.
Babies take a nap half an hour after you named it, and wake up 5 minutes after.
Toddlers wake up at 8 AM or 9 AM, depending which toddler you have.
Teens wake up at 9 AM or 10 AM, depending which teen you have.
Adults wake up at 8 AM, 9 AM, or 10 AM, depending which adult you have.
Which character do you currently have?