Disclaimer: I own neither Sherlock or Doctor Who, nor any of its characters.
Mycroft was at the age where he disliked his little brother following him everywhere and trying to seize his attention. All Sherlock cared about, though, was that his older brother had suddenly stopped agreeing to play with him, which was always why he enjoyed his babysitter coming over- she would always play with him, and she was much older than Mycroft (and much more fun).
"Let's be pirates!" Sherlock said, in the loud tone with which all small children speak.
"But you always want to be pirates," the teenaged babysitter remarked with an amused smile. "Wouldn't you rather pretend to be something else for a change?"
Sherlock thought for a moment; the idea had never occurred to him. "Like what?" he asked.
"Well..." she answered, moving to sit on the nearby sofa with a plop. "We can be astronauts, or cowboys, or aliens from another universe-"
"Aliens aren't real," Sherlock interjected, moving to sit beside her.
"Oh yes they are," she told him matter-of-factly. "I know I've met one for sure."
"You're lying," he said. "Like when you told me it would freeze in place if I made funny faces for too long."
"It's true," she said. "My dad confirmed it, and I can always trust him."
"I like your dad," Sherlock declared. "He never ruffles my hair like all the other adults do." Then he added, "So what did the alien look like?"
"Oh," she remarked, "he was the loveliest man I ever met." She then noticed the way Sherlock was looking at her. "No, I don't mean like that. He called himself the Doctor; he looked like a regular person, but you could tell he was different, like there was something magic about him."
"Will you meet him again?" he inquired.
"I don't know," she said sadly. "This was years ago already; I was your age, actually. And he's gone away now, traveling the universe. But I'd give anything to go on just one adventure with him, to even just meet him again..." She let out a heavy sigh. "I wonder if he'd recognize me?"
By now, Sherlock was leaning against her and she had an arm wrapped around him. They sat in silence for a moment before the boy finally looked up at her. "I don't feel like playing anymore. I'd rather you tell me a story instead."
"What shall it be about?" she asked, indulging him.
"About what it'll be like on your adventures with the Doctor."
"Okay," she smiled, though there was a sadness in her eyes.
"Kate?" he asked her.
"Does anyone else know about him?" he questioned, leaning down to lie his head in her lap, and she began to stroke his hair. "Besides your dad, I mean."
"No," she told him. "It's our secret."
"Good," he replied. He added after a moment, "And when you're with him, can I come along too some time?"
"Of course," Kate replied. "It'll be just the three of us: you, me, and the Doctor."