I used to fly frequently with my oldest 2 when they were very little (from Stansted). There is a family lane, ask when you get there.
As to the double buggy. Are you ok with it being flung onto the plane? Without a bag? If so, then my tips for bringing a buggy are: do not pack anything loose in it. Everything will have to be taken out at security (including rain covers, toys, seat insert things), so make sure that anything in or on the buggy is very easily removable.
Remember, you will have to fold the pushchair at some stage yourself with the children ‘loose’, so use any moment when they are sufficiently distracted to do this.
Carry a small bum bag/cross body bag with: phone, passports, some loose change and your bank card. It saves any faffing for these things at the airport.
At security, everyone will have to be out of prams/slings. Have your liquids out, don’t wear anything with a belt, pack watches or anything you are removing into a jacket packet so it’s not floating loosely in the little basket thing.
You can get a small roll up chalk board (I think it’s crayola) I used to pack it in my bag for the airport.
All nappies changed before the plane.
On the plane, I would have a small bag (I used to use the smallest M&S bag for life), that I kept on my seat with a spare nappy, wipes, and zip lock bags.
Each child had their own ziplock bag with snacks and a smaller bag inside it for rubbish. This bag would be bull dog clipped to the table to easily deal with rubbish.
I never used tablets or iPads. They are more hassle than they are worth with taking them off for take off and landing. Stickers, books, games were better. Constant engagement is required and it’s tiring!
It’s also worth noting that, we have found cabin crew much more hands off since covid. When we fly in Sept, my husband got shouted at for walking our baby up and down the aisle and I got shouted at for queuing to bring the middle child to the toilet.
Also, husband recently flew from Stansted, he said it was very busy with big queues for eateries, donuts worth bearing that in mind too.
Best of luck!