^Regarding paying savings first - I know this is a common suggestion but it's never really worked for me because life gets in the way. I pay into savings regularly but the amount is hugely variable. In April, I get an annual bonus - a couple of treats but can chuck £3k in savings in one month. One off expenses, trips etc - monthly savings vary from £100 - £1000 a month.
In August, zero saved - £500 school uniform and holiday spends, it all goes^
I think the aim is to get into a position where you've already saved up the money for those sorts of things before you need to spend it. Obviously you can't get there straight away, so it's something you need to build up to.
For example, you should be thinking about the cost of next summer's holiday and big school uniform replacement now and put money aside for each month from now on. Likewise Christmas, car insurance, servicing, MOT, repairs etc.
It's also important to recognise the difference between short and long term savings and those that have a defined purpose and those which are truely spare money.
For example, money saved for school uniforms, holidays, car stuff, Christmas etc, isn't really savings, it's more like a cashflow management exercise, you know you will need to spend it over the next few months, so it doesn't build up like real savings do, where the pot increases over time and you will be paying in without taking out, for years to come.
OP, your plan looks quite good, especially if you can cover some or all of normal ups and downs in outgoings (like Christmas, holidays, car MOT, insurance MOT etc) without dipping into your saving too much? Because out of savings, you've also got to account for any major household costs like a new roof, boiler etc, and car replacement, plus saving for loss of income or retirement.
Perhaps earmark some of the savings as short term savings/cashflow and others as longer term savings that you don't exepect to spend on anything except bigger purchases, illness, job loss or as part of your retirement income?