English city break?

[VashtaNerada]

Looking for inspiration for a weekend away with children aged 14 & 10. Museums, history etc! Ideally not too much of a drive from London.

Bath

Or Canterbury

Mine liked York.

Cambridge

York
Long drive, but only just over 2 hrs on the train.

Bath, Bristol or Oxford.

Train to York

I live in Bath and was also going to suggest Bristol or York. I'll throw in Salisbury if they've an interest in the Magna Carta and/or Stonehenge.

Not at all what you asked for but I took my teen to Milton Keynes just before the first lockdown. The primary reason was to give her a couple of ski lessons on their indoor slope (there is one of those iFly places as well). Bletchley Park and the Computing Museum are a short train ride/drive away. As is Milton Keynes Museum with its concrete cows.
Overnight accommodation was really reasonable.

If this is trip is planned for the warmer months, how about heading down to Brighton or Portsmouth?

Bath or Cardiff

Bristol or York via train.

York by train. We stayed in the Novotel there some years back and it was fine. The railway museum was a highlight for us.

Bristol or Bath (I live between the two cities).
Chester
York
Portsmouth

Portsmouth naval museums by train for a day trip.
York by train for the weekend

Another nod for Bath or chester on the train