Oxford or Hampton Court? Or....?

I haven't been to Hampton Court, but Oxford offers a lot of variety. I don't know how the transportation time compares. I myself don't care for palace/castle furnishings, but that is obviously a matter of taste.

The Pitt-Rivers Museum in Oxford has an exceptional collection of artifacts from non-European countries, including shrunken heads. The lighting level is a bit low for my (old) eyes, but the shop loans out flashlights to help with reading the labels.

Hampton Court.

The gardens are fun, there’s a maze to explore and lots of buildings to take a look at. Plus space to run around.

Oxford. Busy tourist-packed beautiful city. Lots of lovely old colleges and Harry Potter connections. Plus shopping, if your teenage kids are inclined that way.

Haven’t been to Oxford yet (will be spending a week there in October), but Hampton Court is a wonderful place. Even if your teens aren’t taken with the idea of seeing Henry VIII’s and Queen Elizabeth I’s palace, actually seeing the place might strike their fancy. We didn’t even make it to the Georgian parts of the palace grounds, but the Tudor kitchens with the enormous roasting ovens, Fish Alley put behind the kitchens, the banqueting hall with the original eavesdropper figures in the eaves, stained glass windows, tapestries, the supposedly haunted hallway where Henry’s teenage wife Catherine Howard ran screaming and pleading for her life, all might make for some mighty fine exploring. Get on the guided tour to make the most of your visit,

Nearby is Kew Gardens, with marvelous roses, a Victorian greenhouse and giant lily pads, and a collection of carnivorous plants that your kids might find ghoulishly fun!

You said Or.....

As much as I loved both Oxford and Hampton Court I’m not a teen

So trying to engage a teens interest I’d suggest taking them to Brick Lane and Spitafields Market.

Maybe do a street art/Graffiti tour.

London Walks Ghost or Jack the Ripper tours.

Carnaby Street or Camden Lock Market for shopping.

Grant Museum Of Zoology

Are they adventurous? See if they’d like to do the climb over the O2 building.

Shoreditch Box Park

Ride bikes

Olympic Stadium

Theatre ( see what’s on at the half price ticket booth. You can look online. Have them pick )

Check Timeout London online for pop ups, events, etc.

An alternative: Bletchley Park if they are at all interested in technology or computers. It would give you your history fix, and there is plenty of room on the grounds for teens to be active. The exhibits are well done. Simply take the train to Bletchley. You walk just a short distrance from the train station and you are there.

Of course, we’re all trying to second-guess what the teenagers are like.

Are they 13, 14, with boundless energy wanting to let off steam? Or languid 18, 19 year olds who don’t get up before midday and are unimpressed by anything?

Or, or, or...

We chose Oxford over Hampton Court Palace for our trip at the end of the month. Sooo tough but my kids (who are a little older than yours) are interested in a "college town" ;) Christ Church for the Harry Potter dining hall, punting on the Thames and we're doing a guided bike tour. Here are some other ideas from my Oxford list:

Lunch:
Wig and Pen
The Eagle and the Child, pub of J.R.R. Tolkien, C.S. Lewis, midday - 10:30 pm
Turf Tavern 11am-11pm
Quod Brasserie & Bar, pizza/burgers
The Grand Cafe for cake and tea (can order scones and clotted cream and tea, not fancy; Was first coffee house in England from 1650)
Sights:
1:30 pm-5 pm Climb 127 steps tower (Built in 1280) of University Church of St Mary the Virgin, $3/$4

Bodleian Library/Radcliffe Camera
Ashmolean Museum, open 10am–5pm
Bridge of Sighs
Free Oxford Walking Tour: https://footprints-tours.com/2-hour-free-oxford-walking-tour/
10am, 11am, 12.30pm, 2pm & 3:30pm
OR Oxford and City Tour, $16.99, 11am & 2pm: https://footprints-tours.com/oxford-university-tour/
Christ Church tour 2-5 pm, (buy ahead) $8/$7
The Christ Church bells chime 101 times at 9.05pm each day, to signify calling in the 101 original students at five minutes west of Greenwich.
Divinity School: first classroom,
Bodleian Library: Treasures Gallery - Shakespeare First Folio, Copy of Magna Carta, original of Handel’s Messiah
Magdelian College (prettiest)
Alice’s Shop, 10.30am to 5.00pm

Pub on the River:
Perch at Binsey
Head of the River
Trout Inn