I don't remember how old my first was when she got to sleep with a blanket the first time. She is 4 now and likes to have two blankets.
We were given a handful of blankets at the baby shower and shortly after before our second was born and I threw the one we liked the most in the hospital bag. Ali had a blanket from day one. She was one of those babies that if she wasn't red from being hot she wasn't happy for the first two weeks. In the hospital I used one of the receiving blankets to swaddle her than I took the blanket that we brought with us to wrap around her. The nurses never said anything about not using it. They all really liked the blanket. She would not sleep laying on her back or laying flat at all for the first 5 months so she slept on her side in the glider. She is now going on 7 months old and she will sleep in the cradle but she prefers the glider still.
I am not as worried about the blankets when she sleeps in the glider because they will fall off her and end up on the floor when she moves because she tends to kick her feet in her sleep. In the cradle I worry about it more but she wakes up cold if she doesn't have at least one blanket. Our bedroom is small and when we have the air conditioning running or we have it off but the furnace isn't running yet it can get really cold in our room at night. Before our central air broke I would put her in a long sleeve onesie, wrap her in the swaddle me wrap leaving her arms out, and put her in a fleece sleep sack and she has woken up about an hour after going to sleep because she didn't have a blanket. I snuggled her back to sleep laid her back down and put the blanket back on her and she slept the rest of the night. Ali is a snuggler. She would snuggle the towel that I used as a burp rag for the longest time. When I would hold her she would be snuggling the towel. She would fall asleep snuggling the towel so I would lay her down for a nap and she would keep the towel because she was hugging it. I just tuck the blankets tight around her and keep them down away from her face.