I'm an independent school teacher.
Our fees go up every year and the VAT introduction is going to be a big hike on top of that. If your child is only 3 now, I would imagine the average cost of a year of private school is going to be a good third higher than you're imagining by the time she's going to secondary school.
On top of the fees, we charge for any residential trips - on average about £2k per trip. Most year groups have a couple on offer every year, and while they aren't compulsory, there is pressure put on parents by the kids to go. There is also a charge for the laptops, for any other equipment needed for science/maths, PTA fees, after school wrap around, clubs, etc ....it all adds up. So you need to budget for that too.
As others have said, you need to think about your reasons for going down the independent route and where your money will be best spent, as that £100k in savings will dwindle pretty quickly and you will be a slave to those fees for a very long time. Why do you think an independent education is so important for your child?
Not all independent schools are equal and not all will give you the advantages you may be looking for. I would agree with others who say that primary is often not worth the money if you are in a catchment area for a good primary school. Secondary is where the money can be really well spent - if you feel the local state school can't give them what they need. You will get smaller class sizes and probably some great extracurriculars. Don't bank on getting better teaching, though. I work with a lot of colleagues who really aren't great teachers but they've been at the school so long that they're too expensive to get rid of. There is a lot of dead wood in the independent sector.
Honestly, having worked in state and independent, and having been state educated myself, I wouldn't send my own children to an independent school unless I had a really particular reason for believing my child would be better off in that specific environment. I had an excellent state education that I would say was academically stronger than the education any of the children have had in the independents I've worked in.
Think carefully about your priorities. You could be spending upwards of £400k on fees over your daughter's time in school - with that, depending on where you are in the country, your daughter could buy a first home when she's 21. Think carefully about what else your daughter could potentially have/do with the money you'd spend on those school fees over the course of her school years, and would that be better for her in the long run, over and above what might not be any better an education than she would have received in your local state school? That's the key question you have to consider here.