As others have said---really important to sit down with the parents and make sure they really really really understand just how expensive this is going to be. Because you do NOT want to pull them halfway through if they are happy.
A generation ago, a comfortably off but not super rich pairsay, a doctor and his stay at home wifecould probably put a couple of kids through private school without too much financial hardship. These days it's not like that at all. The prices have soared, and there is no sign that this trend is being reversed or slackening off. It's possible that your parents in law have not really understood this, because they are thinking back to their day, the it it was not really THAT big a deal to go private.
In the meantime, uni fees have soared too, and the rising cost of housing mean that most people want to be able to help their kids out with purchasing a house or flat as well.
Sit down with them and look at the fee structure of the schools in generalnot just reception age, but right up to age 18 (there will be BIG differences). Then get them to read the private school/fee inflation threads here so that they have a clear understanding and how and why fees tend to rise over timenot just for different age groups but also from year to year, due to things like rising pension contributions for teachers who are living longer and longer.
If they are pretty wealthy and are still comfortable doing this, you can certainly consider taking them up on this, and your kids will most likely do well.
I would be the last person to deny that on average, private schools do generally provide a certain amount of "extra added value" over and above a good state school--not to mention the convenience of getting academics, extracurriculars and so on all coming together in a single reassuring "package."
But calculate the total sums needed and get as much money as possible earmarked or put away somehow. A trust, a designated account, or whatever.
And if the grandparents start going "hmmmm....." when they see the figures on how much this is going to cost them in total, talk to them about doing some educational enrichment for the kids for the moment, and putting some money away for a possible move to private between 11 and 16, depending on how your secondary options work out.
The "gap" between average private and average state can almost certainly be plugged in cheaper and more frugal ways. That is what most of my middle class friends in the UK have plumped here--including those who were privately educated themselves.