Warwick Castle/Stratford Upon Avon?

I think you are correct to dismiss the tour company option
You can get a Family Ticket for the four of you to Stratford upon Avon from Marylebone which is at present £54.80 (may be increased by time of your journey) This does allow a break journey at Warwick on return trip This service is operated by Chiltern Railways. However i think you need to choose between the two towns as there is enough in both to fill a day. The fare to Warwick is the same
Suggest you have a look at www.seat61.com for information on rail travel in UK

Trying to do both Warwick and Stratford upon Avon in one day is going to short change both of them as well as making it a long day.

You need to allow several hours for Warwick Castle. The church is also well worth visiting with the family tombs from WArwick Castle There is also the Lord Leycester Hospital.

In Startford, the thing that may really appeal to the kids is the MAD (Mechanical Art & Design) Museum. Read the blub about it on Trip Advisor. My grandsons loved it and can't wait to go back. They also enjoyed Shakespeare's School Room too (more so than his birthplace). Make sure you go on the chain ferry across the river - a unique experience!

Warwick Castle, Stratford, PLUS a little of the Cotswolds and a little of Oxford in a 10 or 11 hour tour

wow. I won't say that's impossible to do in a day - I mean who defines what "a little" of the Cotswolds and "a little" of Oxford actually mean, but I wouldn't touch them with a barge pole.

Warwick Castle is an almost perfect match for the kiddoes, and we enjoyed it as adults. But it is expensive and to get your money's worth you will want to be there for several hours. The centre of Warwick is a lovely town, with beautiful buildings, a riverside garden (the Mill Garden - the River Avon flows past both the garden and Warwick Castle), a market with a 17th century building, Lord Leycester Hospital (14th Century building, not actually a hospital, closed for renovation until summer, look at pics at their website https://www.lordleycester.com/) and the magnificent St Mary's Church, visible for miles around, with a tower that can be climbed and beautiful bells, and a real view from up top, to mention just a few.

Stratford upon Avon isn't far from Warwick and has various Shakespeare related properties beyond the highly advertised ones including a very nice riverside church (Holy Trinity) that he was baptised in and buried at (along with his wife and other family members), the actual school he went to, and more. Then there's the river, and the canal, and the theatre complex, and plenty more including a butterfly farm https://www.butterflyfarm.co.uk/attraction/

Oxford, depending on what you want to see, can take an hour or a day or more.

The Cotswolds, again, take time to drive around and time to get out and feel the villages, to say nothing of a Tea or lunch, for me is a minimum of a day.

Have the kiddoes seen anything they really want to do?

<> EDIT - started to write this over 2 hours ago. Interesting I focussed on some of the same things

We did Stratford Upon Avon as a day trip from Chipping Campden and spent about 6- 7 hours in that city. We did the full Shakespeare tour that starts from the Bard's parents house where he was born, including a museum, then the place where his house was located (torn down, now a park), the church where his tomb is located and Anne Hathaway's house.

After we left Chipping Campden, we moved on to Warwick, staying there and spending a day at the Castle. We probably spent 5-6 hours there. The Castle is huge and tour extensive as well as the gardens. It is quite a draw for families with children.

Trying to take in a the Cotswolds, where we spent 2 full days touring, as well as Stratford and Warwick in one long day is WAY too much.

First, you won't be able to fully taken in any of the sites,
Second, you and your kids would be dead tired.

The kids will love the Castle and want to stay even longer.

Thanks for all the suggestions! The more I read and watch, etc, I’m leaning towards doing a day trip to Stratford. I’m sure I’ll waffle back and forth 10 time at least, but really appreciate all of your insightful responses!

Frankly, I'd recommend Warwick/Warwick Castle over Stratford. So much to see and do there, LOTS of stuff to interest kids of the ages you stated. The trebuchet spectacle, the falconry show, etc, etc, etc. We had afternoon tea in the Conservatory Tea House and watched the peacocks wandering around the garden... sublime!
We were actually a bit disappointed by Stratford... it felt cheaply commercialized, and if the kids aren't particularly interested in Shakespeare, it may be a bit of a bust for them. (Warwick Castle is also "commercialized" but in a higher quality way.)
Definitely stay away from a day trip by a tour company. They offer the ability to "see" a lot of sites but very little opportunity to "do" the sites.

I am always surprised when I read posts that Stratford Upon Avon is too commercial or disappointing.
We enjoyed the city, especially the Shakespeare tour of four places (house he was born with great museum, location of his house, church were he was buried and Anne Hathaway's house. Of course, I am a big fan of Shakespeare.

Warwick Castle was interesting, but moderately so and the theme park for kids was OK. I did enjoy going to downtown Warwick as well.

I can see by all the great responses that my indecision regarding these two is appropriate. Sounds like “thumbs up” and “thumbs down” for both of them! I’m not particularly fussed either way, so I think I’m going to let my kids read up and watch videos and they can make the final decision.

I've been to Stratford half a dozen times since 1972. I guess seeing the changes (not for the better, IMHO) over the years contributed to my disappointment the last time I went in 2015. And I am an enthusiastic Shakespeare fan.
The one and only time I went to Warwick Castle (2019), I had nothing to compare it to and went fully aware of its "commercialization" and "theme park" complaints. Nevertheless, I considered it an interesting and fun day out.