Is he ready for this?

I don't think he's being immature, everyone deals with death differently. And you said he's lived with his grandma for a very long time, so I would imagine that he was very close to her and for her to pass away unexpectedly would make it even harder to handle. It would be like losing a mother. As his girlfriend you should be there for him to talk to and support him while he's going through this. Just wait until you can tell that he seems to be getting over it to ask him. You guys should be able to tell each other anything so just explain to him why you'd move in. Just talk it out, and if he isn't ready, you can wait and bring it up another time. Don't try to hurry him through his grieving because it is a hard thing to deal with. Hope this helped.