It's been my experience when travelling with Easyjet that their policy is only as good as the people at the boarding desk. We've flown with them at least ten times in the last few years and the implementation of the policy varied a lot. Sometimes they didn't bother announcing the boarding groups and it was a free for all, other times they did announce the groups, but just let people through the gate regardless of the group on their boarding card, on just a few occasions they made people stand back if their card didn't match the group called. Unfortunately, this inconsistency means people continue to chance it. It helps if you are aware of the policy and make sure that you are in the right place at the right time.
Having said that, generally things are run quite well, the flights usually make up any time lost on the departure time, and their planes are more comfortable than many charter planes. We tend to eat in the airport as a way to fill up the waiting time, and just have a coffee on the plane.
If you really want to, you can add speedy boarding after you book, but you should be able to get seats together without it. If you can check in online, or alternatively get to the airport early enough not to be in the last group boarding, you shouldn't have any problem. Get to the gate on time - they usually start boarding about half an hour before scheduled time of departure (having a seat close to the actual gate in the departure lounge makes it quicker and easier to bag a place in the queue when (if) boarding is announced. If you're put on a bus to the plane, sit or stand next to the door and run like the clappers when it stops. If one adult can handle the kids, then the other can go ahead and bag seats.
Failing all that, I've found that people are reasonably considerate where children are concerned and if you find yourself with only single seats left, look pitiful and ask the crew if they can sort something out for you. I've seen it happen.
BUT don't worry about it, just go with the flow and enjoy yourself. If you get stressed out, the kids will too (and that's the last thing you want).
Lucy