Traveling in England in April with 2 kids

Day 1: I suggest you take the train to Oxford and rent the car there. Buy the tickets 11-12 weeks ahead of the travel date on the National Rail website to get the low-priced Advance tickets.

Yes, car parking in Oxford is notoriously limited and expensive, unless you use one of the Park and Rides and get the bus into the centre. I would travel by Oxford by train and rent the car after the walking tour.

So what would we do with our luggage while touring around Oxford?

Woodstock is only 8 miles from Oxford. Go there 1st.

On day 3, you surely would not hang around Thornbury Castle Hotel for a whole day? My advice would be to drive on back roads to the M48 Severn Bridge & leave M48 for A466 at first junction on Welsh side of estuary - then drive into Chepstow to see the medieval castle. (Look closely at the road pattern - which is complicated in Chepstow). Then leave Chepstow & take A466 going N. Just after St. Arvans - a single lane road comes off the AA66 and about 3 minutes drive it has a sharp corner with a rough parking area on R. Park up & follow trail up through the woods for 17 minutes & turn R where it levels off to see Eagle’s Nest Viewpoint. Then back to car & continue up A466 to see ruins of TINTERN ABBEY.

The A466 continues to be scenic going N up the Wye Valley. Yet Rock viewpoint is also well worth a visit or alternatively head west at Monmouth on A40 to Raglan and then west towards Abergavenny. Go up steep hill to Blaeavon and take kids town the Coal Mine at BIG PIT. (Free). On returning to Severn Bridge - go to Raglan - see Castle and then down A449 expressway towards Newport and then east on M4 & M48 to cross the estuary back to England.
Map of bridges:>https://www.google.co.uk/maps/@51.6706774,-2.6865857,12z/data=!5m1!1e1

Map showing Monmouth to Raglan to Blaenavon for Big Pit:>https://www.google.co.uk/maps/@51.7934221,-2.9681038,13254m/data=!3m1!1e3!5m1!1e1

PS. If you wanted to take the train to/from London(Paddington) to Thornbury - you could do it in around 1 hour as most of the line is fast = 125 mph to BRISTOL PARKWAY station. Then hire/ditch car if possible. Advance book specific train for cheapest fares around 11 weeks ahead. www.nationalrail.co.uk
These are the trains (green) :> https://www.youtube.com/watch?v=CGXboGzZiXI

If driving from Thornbury to Bath - best to use Lansdowne Park & Ride. (A P&R is where you park the car on the edge of a city and shuttle buses take you downtown. They have a similar thing at Oxford. Remember, these are old places built long before we had cars.

Blenheim Palace is wonderful and as well as the Palace itself, there are extensive grounds as well as a miniature train to the Pleasure Gardens with its adventure playground (great for getting rid of excess energy) and other kid based attractions. You could easily spend the full day here. A morning just isn't long enough (especially as it isn't cheap to get in either). You won't have a lot of time to drive round the Cotswolds as you may have to restrict this to drive to Thornbury Castle.

I must admit I was also wondering what you are intending to do all day at Thornbury Castle. Why not use that day to visit BAth?

Then head to Windsor on Day 4 and do Legoland.

Blenheim Palace, as I remember, doesn't open early. We spent about 2.5 hours there and didn't depart until well after lunch. Not sure how much you will see of the Cotswolds with that short time left.
We spent a day and a half doing the Cotswolds.

You need the better part of a full day to see Bath, but the Roman Bath Museum is great.
Windsor Castle is great.

I used to live in Thornbury and while it’s a lovely market town to live in there’s nothing to see there besides shops. You could use it though as a base to visit nearby Bristol (to the south) which has the SS Great Britain among other attractions. Alternatively you could drive north to Gloucester, whose cathedral was used as a filming location for Harry Potter. It also has Beatrix Potter’s The Tailor of Gloucester museum and the docks. A previous poster’s suggestion to go over the Severn Bridge to Chepstow and neighbouring areas is also a good alternative. Another option is a visit to the Slimbridge Wetland Centre which is on the A38 north of Thornbury.

When we stayed at Thornbury Castle we arrived just before noon, we were there to meet my mother in law and have lunch with her (which we had at a nearby pub). We then spent the afternoon having a guided tour of the house and grounds and then back to our accommodation (The Gatehouse) for tea. We then had an hour or so to get ready for dinner at the castle restaurant.

If it was my first time in the area then I wouldn't have spent so long at Thornbury but as we have been to that part of the country many times and explored much of what there is to explore we treated our stay as more of a relaxing break.

Thornbury Castle is a great place to stay and worthy of exploration however it would be difficult to justify not seeing what the local area has to offer. I like James' suggestion of Chepstow Castle, Tintern Abbey, Symonds Yat etc.