DuncanRich

Firstly, congratulations.
The amount you need to put aside will depend on the timescale available between now and when you first want to start sending your child to a private school. It will also depend on the rate of return that you can achieve on the investment, which in turn will, to an extent, be dictated by the amount of risk you are prepared to take.
There are many investment options available to consider and you will want to use the most tax efficient ones for your circumstances first, so this will normally be an ISA. You and your partner can each put £10,200 per annum into an ISA. After using your ISA allowances in full each year it will then depend on your personal situations. For example, if you do not use your Capital Gains Tax allowance in full each year then it makes sense to aim to do that.
As a new father you will also need to consider where the money would come from for this private education if you were no longer around to make the savings and investments.
a full, comprehensive, and integrated financial planning assessment is what you really need now.

Well really I am afraid there is no substitution for regular substantial payments, however there are a number of things you can do to improve the potential outcome.

You should sit down and think about the following issues;

What do you want from a pension (how much, when)?

How 'active' do you want to be in managing it (there is no substitute for involvement in how your money grows for making all concerned concentrate on the outcome)?

Look hard at the costs - cheapest won't necessarily be best, but make sure that what you are paying offers good value.

Get good advice - and at risk of showing my prejudices I'd suggest an IFA might be a good place to start - they will not be cheapest, but ask them to demonstrate that they can add more value than the differential in cost, a good IFA will be comfortable with that discussion.

If you have surplus capital, or annual bonuses or profits surplus to your immediate needs a lump sum contribution can be a major boost, but you may wish to think about phasing how it is invested, so that you aren't exposed to any sudden market drops just after investment.

Decide how much risk you can take - classic human psychology is to invest at the top of a bubble, and sell at the bottom - the exact opposite of rational behaviour - of course this is only obvious with hindsight, and by ensuring you are not in a market which you will not be able to stand when things go wrong then you can hopefully avoid this dreadful mistake. Getting timing right is very very difficult, getting it wrong is very very simple, so the best option for most people is to ignore timing, and the only way to do that is to be in markets where getting it wrong won't force you to disinvest for emotional reasons.

Oh, finally, and very importantly, if you are employed check whether your employer will contribute on your behalf, either into a scheme they sponsor, or yours - for many people the largest part of their pension contribution comes from their employer - joining an employer funded pension is really like a payrise - it may 'cost' a few pounds from your net salary for your contribution but the benefit is usually comparatively huge.

Best of luck - and remember the worst decision you can make is to do nothing.