Juat done long haul (though not quite as long haul) with a fussy young child and am vegan.
Order your vegan meal before hand.
Most airlines also have kids meals you cna order as a special meal which are generally more basic.
You can pack up a lunch for the child too. Like a lunchbox you'd take out with you. We had a bagel and pom bears, cucumber and fruit.
As its 14 hours plan a second meal. We did for 9 hours as it would have been her tea time and a hangry child going through immigration is not fun. We cooked up pasta and grated cheese and took it cold. Along with some saldy bits.
I kept it all cool on the way to the airport in small coolbags and took mini freezer packs i put in clear liquid bags for secruity. Note heathrow said they cant accept ice but let them through as they'd defrosted by the time we got there. But were still cold so did their job. Remember loquid wiae you can only take 1 clear bag per person of liquids up to 100ml. So you can take those sucky yoghurts (85ml) if you put them in a clear plastic bag.
Take all the snacks for you and them. Depending on the airline vegan food offerings can be interesting and might not appeal. So have something to keep you going. I'm always cautious of nut products just because i've previously been on flights where we've been asked to regrain from eating them due to a severe allergy on board. So then you're stuck with snacks you cant eat.
As others have said check food restrictions for landing. You can leave stuff on the plane. I took a small carrier bag i kept rubbish in throughout the flight. Made it easier. Though only country i've been to that i'd say cares is Australia. An apple there is like you've brought a bomb into the country. Other places may say they dont accept x y z (usua3llg meats and cheeses and things with soil on) but i've never been stopped and i've travelled alot.
Only thing i buy at the airport is water or i take .abottle and fill out own.
Every major airport in the uk has somewhere to eat before you board with vegan options too.