I found this place for you. Not central London, but Greenwich is such a fantastic place to take children anyway - there's so much to see and do. Never been to this place for tea but it looks lovely so think we'll try it out too.
How about the Wallace Collection - it's a museum in Manchester Square with a fab atrium out the back with a restaurant in it. Here
You can have afternoon tea, but can also just have tea and cake or icecream. The museum itself is stunning, a real eclectic mix of stuff, from suits of armour to Catherine the Great's ice cream coolers! And if it's not really your thing, there's another Patisserie Valerie around the corner in Marylebone High Street which is lovely too.