Tiny Humans
Sherlock probably should be insulted. People assume that anything involving sex will baffle and confuse him to the point of rendering him useless. So when John and Mary would hand him their newborn, they would treat him like an imbecile. Support the head. Use the blanket. Don't bring the baby near the chemistry set.
However, when Mary let go of baby John and Sherlock had him in his arms, he showed them that he understood what a baby needed when being held. They were tiny humans, after all, why would it be such a foreign concept to Sherlock? Unlike their slightly older counterparts, newborns were simple and had basic needs. They were nagging, blaring needs - however, they were still basic.
John squirmed, both the baby and the man. "I have him," Sherlock assured them as he walked to his chair and sat down. Briefly, he looked up at Mary and John and offered them a smile. "You two have done well with this one," he said, offering the sentiment more for Mary's sake.
John coughed before saying, "I hope you know you will be his godfather, and you aren't weaseling out of this one."
"Of course, John," Sherlock said, "someone has to teach this child right."
John looked at Mary, his eyes insulted but the smile on his face expressing that he was going to choke his comments down. "Babysit, too?" Mary blurted with a small laugh.
"Of course," he said, smiling at the baby as he smiled back, the visual stimulation pleasing for the infant. Mary gestured for her small victory.
In their circle, Mary and John would never have a lack of people to look over little John. Mrs Hudson had already demanded to be the first to watch the baby if needed and wanted to help Mary get back to work when she felt comfortable. She cited missing hearing the sounds of children in her home and would love to help. Molly offered, no doubt running to her current paramour and scaring him with the discussion of John and Mary's new baby. Lestrade offered, but mostly out of obligation, no one took that man serious.
While Sherlock never offered, the swing next to him begged to differ and the preparations he did with Mary told the couple that he wanted to be there. John pulled Sherlock aside and informed him that he was doing more than his sister would ever bother to do, and he would never forget it. Sherlock waved the gesture away and offered some insult as he assisted Mary with the arrangement of the room. As detached as his voice was and despite the things he never said, his actions and gestures surprised those around him.
John was slightly worried when he found Mary and Sherlock viewing a baby book together and had to establish his place as the father of the baby, but he was grateful Mary had someone who knew her, her past and all, to talk to other than him. In fact, he was even more grateful when Mary would text Sherlock odd requests for food or media. Watching the wheels of ludicrous requests turn on Sherlock was refreshing. Sherlock was oblivious when John tried making the connection for him.
Sitting with John Watson Jr, Sherlock marveled at the couple's creation. The baby instinctively accepted Sherlock and his gentle grip and began to close his eyes, finding comfort in his arms. At that moment, Sherlock could only imagine what could be going through that tiny brain as it begins on its developmental journey. Sherlock could only imagine the man this baby could become with parents like John and Mary - - and a godfather like Sherlock.
The idea came about from reading many stories about the bumbling idiot known as Sherlock around babies/children.