Well done for posting about this and tackling it.
A few things
Health promotion person I spoke to said that BMI calculation has been discredited, so don't take it as gospel, I haven't researched it further but I guess someone on here can enlighten us. And that doesn't mean you don't need to change things, but there is also an element of him growing out of it. He is only 7
If you are a single person the key to this I feel is as follows
You are probably (and I'm assuming) budgeting and doing stock control of your cupboards going forward.
So the rule is, what you don't buy they can't eat (as it isn't there)
So knock the following off your list right away and you will find it not only improves things but saves money
Chocolate and chocolate spread
Biscuits (right off)
Brioche
Crisps
May seem harsh, but knock them RIGHT off
He will occasionally get these as a treat at parties/sleepovers no doubt, but never, ever have them in the house.
He won't starve and neither will you.
When someone said 'replace processed foods with home cooked foods' they are quite right, however some people imagine that this involves more work and more time.
It does't always, there are lots of healthy snacks that don't involve cooking, then there is batch cooking whatever.
Sure others can come up with more tips.
And work towards a meditarranean diet. Olive oil.
Nothing wrong with beans on toast.
You have already got vegetarian things in there, the options are many. Cook together (or assemble sandwiches together wholemeal bread) it is fun and you can save money on activities that way as well
They can research veggie recipes on internet and you can take it from there.
good luck