Move in newly decorated house with young children

A decent paint is going to cost you which ever brand or type you use. I have in the passed used a paint for a customer in much the same situation as yourself (young children wise), back then it was called Clean Extreme now they call it Super Scrubbable; it's super easy to keep clean but only comes in matt finish. Matt walls / ceilings with satin finish woodwork is the way we went with their work. Have a look on my profile at pictures 1 & 4 to give you an idea of how it looked once finished. It's the same though it wasn't cheap at the time, not sure of the cost now. Below is a link to the paint I mentioned for the walls & maybe do the ceiling in a decent matt emulsion? ( https://www.crownpaints.co.uk/products/hall-stairs/matt ) Other makes & brands are available. I hope that might have been of some help.

If you're really concerned with the health aspect have a look at the Nursary Paint company. They have products made from soya resin that are completely child safe. There is sometimes a compromise with durability but I have used it on woodwork and it's pretty tough. F&B is ok but on walls you want to use Modern emulsion that can be wiped. For wood you can use their full gloss or eggshell - both are tough enough for children and eco enough. If you struggle brushing F&B I would suggest you are using it too think - you pretty much always need to add water to the paint before you use it so it will be a case of finding the right viscosity to get the finish you want. But it will give a very good finish if you can get it right.