We have always paid out of income as I started having babies at 22 and we both always worked full time. Hardest times have been full time nanny for youngest plus 2 school fees for oldest type of periods.
My ex husband (we were still married then) only had to pay 15% of one child's fees but I believe the girls' schools offered teachers virtually no discount so very much depends on the private school. We know a couple where they got 3 children through prep school to age 13 paying just the 15% where husband worked and then to age 18 at a leading boarding school for a similar discount. Makes their 2 teachers' full time wages (plus staff accommodation so they had no mortgage) actually a better deal than people realise teaching can be.
Then my son got a music scholarship at 12 so that helped with a discount on the fees. Also when I was in my 20s paying school fees we did things like holiday at Butlins, very cheap shopping bill, all uniform second hand etc etc.
With the twins I had more money so it wasn't so much of an issue. Our divorce order says whoever the children live with I pay the school and university fees... lucky me.
By the way although occasionally I have found grandparents paying when I was at a private school and a few since we certainly didn't find loads doing that around here.
On whether you invest and pay out of that depends when you have your children and what you afford. I think my brother who is starting all this with his wife over 40 has had 20 years or something to get ready. I had zero and they do have savings if they have to use them to fund some fees.
I think the suggestion of extra money earning is good too below. We have done all kinds of thing - teacher husband working on other job in school holidays, private teaching, me writing books, marking exam papers etc etc Today is Sunday. I am working.