"The concern you should have is what would happen if your husband lost his job - ... - you need savings in case this happens."
Indeed. While DH seems fixated on private schooling (and I went to boarding school for a while before going to a local grammar school myself, so know the difference between classes of 18 and 30), have you considered how much you will need to be earning when 2 are at university and 2 are teenagers ?
My guess is that long hours and these financial commitments could affect your husband's health such that he perhaps would have to take time off through stress, yet there's almost no way to continue with your lifestyle if it happened.
I'm guessing that private schooling and other cost will only go up, with all the things to "keep up with" peers that your children will have, as SugarplumMary at 17:18 suggested.
How well are the teenagers doing at school ? Where are your local state schools in any league tables ? Would they benefit from seeing how ordinary folk do at school, or are they too close to exams ?
If they've had a good education all through to their teens, they should be confident and capable if they didn't continue at private schools, but would they fit in, even, at a state school (or be bullied because of a 'posh' accent {not wanting to touch on stereotypes, but hope it can be seen as a valid point}). In case you are wondering, am just exploring what would happen if the world came tumbling down (your DH in hospital, or out of work, for 6-12 months)...
There's no 'slack' as you spend nearly everything /save for children / boost DH's pension, but what if he left you ? (OK, worst case scenario, and hope it never happens!) As others have indicated, no savings = bad news.
Admit I have no savings, but also just me to worry about (and only me to do the worrying).