England/Scotland/Ireland in 12 days

The only issues I can see with doing a land tour company would be that you are locked into a schedule which doesn't allow a lot of flexibility. If you truly have a total of 12 full days (i.e. not including travel days) I would recommend sticking to England and possibly Scotland.
You could, for example see London and Edinburgh - no car required.

I am not well versed in London's Harry Potter offerings, but as I recall, there is a day trip to the studio I think - son did it a couple years ago and thoroughly enjoyed it. Also, London Walks does a walk to locations in London where some of the series was filmed.

I would recommend you get a good guidebook or two and have everyone go through them to determine interests and then, once you've put together a rough itinerary, the good folks here can help you fine tune it.

You will need to do this on your own if you take the 2 year old and the 9 year old. Most tour companies won't take children younger than 10 or 12.

First, I'd drop Ireland for this trip. Too much travel involved to include it, and they don't have the rail /bus infrastructure that England and Scotland do. You could get a nice taste of London, Edinburgh and one other place- possibly Cardiff Wales, or Bath England- in 12 days and move at a slow enough pace to accommodate a 2 year old (though, personally, I think I'd find a family member for one that young to stay with and only take the 9 and 11 year olds). Consider something like this:
Fly into London- spend 4-5 nights and include 1 or 2 days trips by train to any of the following- Windsor (1/2 or full day), Greenwich, Kew Gardens, Dover, Canterbury, or other easy to reach by train locations.
Train to Cardiff or Bath. Spend 1 or 2 nights.
Train to Edinburgh- spend 3- 4 nights. Do a Highlands day tour with tour company on one day.
Fly home from Edinburgh (might have to return to London for one night , depending on destination).

Consider renting an apartment in London and Edinburgh.

Is this 12 days in the UK or 12 days including your transatlantic days ( effectively only 10 days in the UK)? Regardless, if you don't want to be in a different hotel every night, then a bus tour is not for you. Even if you could find one that accepts kids of that age, remember that you are locked into their itinerary and schedule- which may not work well with the kids.

I also agree than you don't have time on this trip for Ireland. Plan to see it on a later visit. I'd recommend picking 2 (or at the most 3) places in England and/or Scotland. At least one of these (?London?) should provide opportunities for day trips by bus or train. Travel between cities by train (buy tickets in advance to save £££ on fares). Look at flying into one city and home from another to reduce time lost due to the need to backtrack.

I strongly recommend not trying to visit all three countries in only 12 days. You will spend way too much time traveling. Why not eliminate at least one country?

If you love Harry Potter, you may enjoy visiting Warwick Castle, north of Oxford.

There are various Harry Potter locations around the country such as Alnwick in Northumberland and Gloucester Cathedral, but personally if you are fans then the Harry Potter Studios tour just outside Watford is well worth it even if it is expensive for a family of four. You can only go with advance tickets and these sell out a few months in advance for weekends but midweek outside school holidays should be easier to secure nearer to your departure date. You'll easily spend about 3-4 hours there. Can be reached by public transport from London. Our train network can be expensive especially if you want a flexible ticket that allows peak time travel. Sometimes flights are cheaper although getting in and out of the airports often adds extra expense. Newcastle has a well connected airport for public transport connections and would be a good option for potentially heading out to Alnwick, Durham or even Scotland. What do you like to do as a family? You could hire a boat and sail upstream on the Thames. We also have a number of nice canals and waterways where you rent a canal boat. I agree if you spend a few days in London (recommended) you probably only want to take in maybe two more centres. Perhaps another city and then something in the countryside or by the coast?

Unlike previous commenters, I won't try to talk you out of all three countries if that is your goal. Also, I think one commenter misunderstood the ages of your children -- they are 9 and 11, is that right?

You could, for example, fly open jaw US-Dublin and London-US. Spend your first 2 nights in Dublin, which will give you enough time to see the highlights of the city and get a taste of Ireland. Fly Dublin-Edinburgh and spend your next 3-4 nights in Edinburgh, including a day trip to see the countryside. Then take the train south, for example to Durham or York for a night or two, and continue by train to London. End up in London for 4-5 nights.