"You won't often refer to your two children by name at the same time anyway.".
Really? Presumably you aren't including the next fifteen or so years in that? My second isn't even three months yet and I call the pair of them "X and Y" all the time, so much so I run it altogether into nearly a single word!
Tilly and Teddy sound like children's entertainers to me. If you have a handful of other children in between so that it's Tilly, Mike, Jane and Teddy, that might be ok. But otherwise I'd avoid.