Start with pre-made Characters
By which I don't mean generic characters that could be in any fantasy story, but by creating a character specifically for each of your children. These characters should be based around the characters from movies and TV shows they each seem to be drawn most toward. It's not the perfect fit that character creation is, but it can let you leap over the hurdles that character creation presents to a new player and let your children get right into the game. Your child's statement about it feeling exactly like a movie is what you want to most emulate.
Use essentials classes only, specifically MBA focused ones
Essentials classes (the Knight, the Slayer, the Warpriest for example) are built around having strong class features (always on) vs. making choices between how to use encounter and daily powers to most benefit the party. They work best in Heroic Tier (levels 1-10) and are strong classes that do not require as much optimization as the AEDU classes do. more importantly they focus on what are iconic archetypes both within and without of D&D. The Knight and the Slayer for example are both actually sub-classes of the traditional fighter. The first wears heavy armor and is all about protecting his friends while the second is about dealing as much damage as possible to monsters.
Use inherent bonuses
One of the best rules options you can take advantage of is inherent bonuses. Found in Dungeon Master's Guide 2, p. 138. as well as the Dark Sun Campaign Guide Book, p. 209 (where it was expanded) inherent bonuses take the place of magic items for the purposes of system math. At set levels the characters will gain +1, +2 etc. to their to hit rolls, damage rolls, and their defense stats. Magic items are still compatible in that their properties, item attacks, and bonuses to crit damage remain, but their mathematical bonus does not stack with inherent bonuses. Whichever bonus is largest is used.
Focus on the story and the adventure, adjudicate their actions to fit
If you were playing 4e with adults I would wholly say to depend upon the use of the powers their characters have and play the system as-is. However this may be a difficult pill for your children to swallow all at once (there are plenty of adults on the internet that can't handle 4e's separation of fluff, the descriptions and lore, from crunch, the hard rules themselves). When your children's turns in combat come up or they are making decisions out of combat ask them what they want their character to do and then based on their answer formulate what their character would do mechanically and then walk them through what their character does. This way you can introduce and have them take over parts of the rules at 1 piece at a time.
I know you're not a native English speaker, but in many circles of general nerd culture in America and on the Internet, there is a phrase for what you're doing right now: your spaghetti is falling out of your pockets.
In all seriousness, though, you seem very anxious about your situation and you should take a step back and relax. You say you're good at improv, but it looks like you're already sweating bullets because your party got sidetracked instead of getting to the quest you wanted to take them on.
I'm going to put my direct answer to your question first and some general advice afterwards.
Your players probably won't miss their characters. Why? Well, you generated them, and even wrote up their backstories. These characters were never made by the players playing them, and while they might have grown attached over a few moments, I find it extremely unlikely that they'll shed tears over a party that lasted for four sessions... that they didn't even make themselves. Even if the characters are extremely cool and interesting, that element of personal attachment just isn't there, which allows the players to truly say that they helped create an exciting story with interesting characters. This obviously varies from person to person, but again, I find it very unlikely that your players would have grown very attached in this case.
Go out with a bang. You know that amazing epic encounter you were saving for the climax of the plot arc? Yeah, run it now. You'll have to make some tweaks because you obviously aren't quite there yet, but get them there as fast as possible. Feel free to kill off PCs or even have a TPK at this point; memorable deaths are often much better than "and then they lived happily ever after."
Here's a few things to keep in mind when starting your next campaign:
Your content will come to light eventually, and it will be good. The quests you've designed will always find a way to come forward. Even if your plot arc is entirely ruined by something the players did, you will be able to recycle the content you made but never ended up playing, and I encourage you to do so for your new campaign. The only things that are truly lost are "hard" materials, like NPC stat sheets, etc.
You're in control, and therefore, you set the tone. Sometimes, it is best to take a page out of Gygax's book; after all, this is your campaign, and you put a lot of effort into it. Obviously you shouldn't take the entire preface from the AD&D DM's Guide to heart, but there is a point where a DM should draw a line in order for there to be some kind of structure, assuming you want your campaign to go anywhere. If your players are goofing around and killing NPCs for no reason, or making light of important people in-character, then they should be ICly punished for it; reprimanded for insulting a nobleman, pursued for attacking innocents, etc. It is also very possible to play a serious game in character and laugh until you're blue in the face out of character. This frequently occurs in the Dark Heresy games that I've played and ran.
You had better get used to murdering your darlings. This is a phrase commonly used amongst writers and creative designers everywhere in the U.S. The phrase means that you'll have to scrap ideas frequently, including ones that you really, really liked, so you had better get used to it. The saying is intended for use in the writing, film, video game, and other industries where a publisher or producer oversees your work, constantly telling you what can and can't make it to the final product based on time and expenses. However, it works just as well for when your ideas can't make it to the game because your players did something insane. And, on a related note...
Plan less. I don't know how much effort you're putting into writing everything ahead of time now, but you might want to ease up on that. From what you're telling me about your role-playing experience, it seems like you've been playing in a "safe" and slow environment where you rarely, if ever, have to scrap or re-do material. This happens literally all the time in regular tabletop RPGs, thanks to the insanely unpredictable nature of 4-5 different people working together. It will save you a lot of anguish if you lay out a basic outline of what's going to happen and then add the details once you're sure the players will be arriving there next session, or maybe two sessions later.
It seems like you've learned a lot already OP, which is great, but scrapping a campaign after 4 sessions (and while your players are all enjoying it) is something you should really avoid. If everyone else is having fun, consider either shaking things up a bit and changing your own notes, or coming up with a way to set them back on track, which doesn't always need to feel contrived or railroad-y.
EDIT: Well, now that SevenSidedDie has made that edit to your post, there are a couple of details that I didn't quite catch before, no offense. Since you said you have a month between each session, it seems like you might be over-planning because you have a lot of time between sessions. Heck, you might even consider having more frequent sessions, if you can't stop yourself from overthinking it in the intervening months. If in-person is not an option, use Skype and/or Roll20.
Best Answer
Your focus: Having Fun!
The only real goal there is when DMing is to make sure everyone at the table (including you) are having fun.
What is Fun?
Well, that's going to be pretty table dependent, but that's also where having a Session Zero (and being the DM for it.) But gathering folks to talk about what they want to get out of it and what you do can definitely help guide you.
And don't be afraid to change course when things aren't working. Being honest and up-front about that is important. I had started up a campaign and heavily focused on difficult combat, but the players were newer and I wasn't sure they were enjoying it. So I talked to them about it and we came up with a plan to more slowly get them into more tactical combat when they had better command of the rules and their characters.
Planning (and lack thereof)
This is another area where you'll get a lot of opinions. Personally, when developing a homebrew world I come up with broad strokes. Large maps, key players and problems, and my overarching theme/story arc. Once the players start interacting, I start improvising a lot. I want them to impact the story as much as I tell it, and that means thinking on my feet. Keeping too strict a story doesn't work for me and that's not generally how I want to DM. But others can, may, and will differ - and that's okay!
You've got some more leeway in planning locations more than PCs as you can better improvise characters than maps - but concentrate on the physical locations they are currently in are very much seem likely to go to. If they go somewhere you're not prepared for, don't worry about it. Keep it off-map until you have time to build a map for it. It's hard not to stress, but it all works out. Players understand you can't plan for every eventuality and as long as the story is fun and the table is having fun, lots of leeway will be given.
Homebrew or published
I haven't played a full published adventure...maybe ever. That's a personal choice and maybe you're in the same box. It's a big step to come up with a big storyline and world. If you aren't comfortable with it, maybe start with a short published adventure to whet your chops and see what you like, what you don't like, and help provide new ideas.
But it's perfectly okay to come up with something entirely on your own.
The key is having fun. As long as everyone is, you're doing it right :)
You may want to limit to just official/standard content to start with until you are comfortable with playtest or 3rd party content. This lets you play within tested rulesets.
Session planning
Just to be clear, I do plan on encounters in a session. But I generally make a couple that I think they may go with and have a couple in my back pocket for things I can use if they've gone in entirely different directions. I try and not railroad too much, but as you tell the story, the 'railroading' may happen as they follow the hooks you've provided.