If there is enough demand for a child-free cafe one will turn up sooner or later. I'd say it's unlikely in SE23 as it would alienate too much of the population but who knows - would be a hell of a risk for someone which is why I don't see it happening. It might occur through more subtle methods, i.e not having high chairs, room for Prams etc, no child menu for food. Ultimately during school term you are only going to have under 5s, some of who will be very young and whose noise volume can't always be controlled.

I am of mixed British and European vintage, and in mainland Europe attitudes can be quite different. In Spain for example, kids are much more allowed to be, well kids. The noise is much louder generally to you don't notice as much...

The best solution may be listening to some music with your coffee with some noise cancelling headphones. I head the sound of fox calls are very soothing