My first UK trip, probable itinerary June/July

I'm not a fan of the Cotswolds either. I've been there on a Mad Max tour from Bath, which also included Avebury and they do a tour that includes Stonehenge and the "house of Harry Potter's parents", if your child is a fan. I would consider adding another night to York-there's lots to do there. There is an easy walk on top of the wall. EDITED: When you're in Cardiff, you need to go to the outdoor museum at St. Fagan's, easily reached by public transportation. It will take the better part of a day.

Please include info regarding interests/gender of your child for more ideas.

So, you are touring Scotland in the middle of your trip (between Bath and London). I’m just curious why you have placed Scotland in the middle instead of putting it on the front end or back end of the trip.

9 different places in 3 weeks feels too many for me. I understand your desire to see as much as possible but you will find yourself spending a lot of time travelling and less time actually seeing stuff.
Your plan is concentrating on cities at the expense of the countryside. Fine if that’s what you want but I would suggest you might want a bot of both.
There is so much to see that you can make a case for extending your stay in pretty much everywhere (except for me Inverness*). I would recommend you choose to drop 2/3 places to make room to slow down. Remember two nights in one place means only one full day. I would spend a few nights in either Liverpool or Manchester and visit both the other and the fine Roman city of Chester.
To be clear re the first post St Fagans is excellent but it’s near Cardiff not York as it might look.
* Inverness is OK as a base for a few sights but it doesn’t have a lot to see. Moreover Loch Ness is one of the more boring lochs in Scotland. Inverness certainly doesn’t deserve 3 nights when many others only have 2. Personally I would drop it altogether.

Is there an All Creatures Great and Small Exhibit in York?
That's only in Thirsk that I know off, a short train ride North of York

Why Birmingham? I'd suggest Chester instead or a couple of days in nirthern Wales.

For that matter, why Cardiff? I like Cardiff but for a first timer I would go to northern Wales instead.

Thanks for the recs! Main reason for having Scotland in the middle is that I fly in and out of London, I could have done open-jaw but since I did not I'd rather spend my last nights in London to take it all in and an easy flight out. While going to Scotland I thought I would stop in some places on the way (plus Rick recommends that way :))

I'll seriously consider Chester, it looks great. I probably will cut down some places, perhaps also Inverness to 2 nights and York 3. I picked Inverness because of, well, I never been there and it's convenient to get to Loch Ness, a place where I think we should see at least once.

One reason I am not going to much countryside places is a lack of a car. I know some buses do go to a few places, still I plan to take at least one or two tours into the countryside, perhaps from York.

I don't know much about North or South Wales, I picked Cardiff because it was the capital, easy to get too and St. Fagans, but Northern does seem nice as well. Any suggestions on where to in Northern Wales, it seems to lack public transport, or at least not as convenient.

My son is 13, he did read all the Harry Potter books twice but any 'Potter' location is not a must see for him. Other than that he is a normal teen, we do go hiking sometimes. Thirsk is the place for All Creatures but could not remember the name but knew it was close to York, thanks :)

BTW I just looked at Rick's 21 day suggestion, has has the traveler sleeping in 10-11 different places in 21 days!

The usual route on this forum from Bath is to go to Conwy in North Wales- which has it's own railway station, although Llandudno Junction (a mile across the bridge) is almost as convenient. From Bath that is typically one change at Newport (Gwent) every 2 hours, but if you've been to Cardiff (for St Fagan's) first Conwy is a direct train every 2 hours (the train you change into at Newport, coming from Bath). Conwy has a great Castle. Many other north wales places are easily accessible by train or bus from Conwy, including Caernarfon Castle. Or from Llandudno Junction train down into Snowdonia- the Slate caverns etc at Blaenau Ffestiniog.
You then get to Liverpool by changing trains at Chester, and there are now direct trains a few times a day from Liverpool to Glasgow (otherwise change at Preston or Wigan every hour).
That's the kind of standard way to progress.

I'm thinking of staying in Chester for 4 nights as a base camp, June 23, 24, 25, 26. Get into Chester and explore. A day trip to Conwy, day trip to Liverpool, a day in Chester, then I'm off. Thoughts?

Sounds good. What we hope for now is no rail strikes round about that time.

You can go by train from Bath to Salisbury in just over 1 hour with hourly trains. From Salisbury, the Stonehenge Shuttle Bus will take you to the stones. On returning to Salisbury, I would walk down to the magnificent medieval cathedral before taking a train back to Bath.

Cardiff is about 1 hour 10 minutes by train (hourly). This city will need at least 2 nights as much of a day will be spent seeing the city centre and Cardiff Castle and the National Museum of Wales plus a visit to the waterfront (Cardiff Bay). St.Fagans Castle & Museum is a 25 minute bus ride west of the city (Cardiff Bus 32) and this needs at least half a day as the site is vast. Caerphilly Castle is a huge medieval fortress built in the 1100’s with trains usually every 15 minutes from Cardiff and the journey takes about 20 minutes.

From Cardiff, I would go by direct train to Conwy - which has a medieval castle. (Up to 2 children can travel free on Transport for Wales trains - as long as they are with at least 1 full fare paying adult). Staying in the Conwy area will give you access to the seaside town of Llandudno & you might also like to consider Bodnant Gardens which can be reached by bus from Llandudno Junction. You can also reach Caernarfon Castle by taking a train to Bangor and then a bus or use buses for entire journey. (Suggest you read some of the posts in the Wales section for more info).

From Conwy, I would go to Chester and have a look around before taking another train into Liverpool for a short stay in that city. From Liverpool, head by train to Glasgow for a stay.

From Glasgow, I would not go direct to Inverness but take the West Highland line to Fort William for accommodation. Fort William lies at the base of Britain's highest Mountain = Ben Nevis. It is also the place from which the steam train featured in Harry Potter heads west down the line to Mallaig on the west coast - crossing the famous Glenfinnan Viaduct on the way. (Normal diesel trains also operate the route). Back to Fort William, you could take a bus NE to Inverness passing Loch Ness on the way. Go to the Scotland forum where experts will give advice on this or taking the ferry from Mallaig to Skye - which has wonderful scenery but I am not sure of public transport. You can get back to the mainland via a bridge to Kyle of Lochalsh which has trains to Inverness.

Google the places mentioned.
https://westcoastrailways.co.uk/jacobite/steam-train-trip

"BTW I just looked at Rick's 21 day suggestion, has has the traveler sleeping in 10-11 different places in 21 days!"
I disagree with Rick as well!

That is a lot of moving around. Rick's tours have everything done for you so it's a lot easier to keep moving when you don't have to worry or think about anything.

Please remember that packing up and checking in and out of hotels is tiring and takes away time from exploring. If it were me I'd cut one or two cities from your itinerary. Sometimes kids like to have downtime or a day just to chill or explore.

I'm defiantly not a fan of quickness of Rick's tours.

I have seen a lot of the UK over the years. I look at Wales, Scotland and Ireland as places to visit on separate trips. We like to pick a spot and stay there for two weeks, branching out from there if you really want to see somewhere. What have we missed by doing this? We don't think about that, because we really get to know an area and it's inhabitants this way. For a teen it might be boring, however.

BTW I just looked at Rick's 21 day suggestion, has has the traveler sleeping in 10-11 different places in 21 days!

but remember that those suggestions are based on what the tour groups do. The tour group has all their transportation waiting for them - instead of the other way around -, the buses have the hold underneath, the buses go right up close to the attraction, driven by professional drivers well familiar with the routes, and all the hotels are ready for them too.

Normal mortals can't easily keep up that pace, and can easily get off plan.

If you're already in Bath the Cotswolds are not far away for a day's trip from Bath, even without a car. However public transport is harder to find and access which is why many opt for cars. But don't write off the Cotswolds as a possible destination by foot.
Chester is a lovely town, however I'm concerned that you're not going to see very much of it and so, other than the consistency of staying in the same place, it might be helpful to ask yourself if you wouldn't benefit more from just staying over in the places you want to have a day trip to and popping by Chester on the way up north. Day trips can also turn out to be more exhausting if you do a series of them in short succession.
You're spending the most amount of time in London, which is fine if you have things to do. However if there's anywhere you are going to day trips from, London might be best given how much time you'll have there anyway.