Help with early planning logistics (London, Bath, York)

Sounds like a nice trip, and certainly doable. I'd go straight to Bath on arrival for a 3 night stay. Day trip to Stonehenge with a guided tour or DIY by train to Salisbury and bus to Stonehenge. Then train to York for 3 nights and then train to London for the remainder of your trip. Or shorten either or both of the first 2 places and add them to London - lots of day trip possibilities from there.

Agree with CJean. Head to Bath upon arrival, spend 3 nights, then York for 3 nights, and finish up in London.

To get to York from Bath we have to travel back to London first, then London to York, correct?

Not necessarily. Look for train options on the nationalrail.co.uk website. There are some trains that change in Bristol.

If you take a train for the 15 minute hop to Bristol (Temple Meads), you can then change for a direct train to York. If you were to go from Bath back to London (Paddington) and then go from London (King’s Cross) to York you would have to cross London and be going up and down the London > York leg twice. So, going via Bristol means you are travelling a triangular shaped route on different tracks.

The other advantage of going via Bristol is that you will only have a change of platform to negotiate with bags rather than changing stations in London by the underground (subway) or taxi.

Once you’ve chosen a route others can advise the best ways to minimise fares

Recently did your itinerary so happy to add our experience. Like you, we flew into LHR then took direct bus to Oxford. After 2 days enjoying Oxford we took train to Bath, a/k/a Bath Spa. Then train from Bath to York via Bristol to avoid station change in London. Finally took direct train from York to London Kings Cross station. You can save some money by getting the Two Together rail card as we did. We booked all the train travel on the Trainline app. You pay a very small booking fee, but I found this site easier to use and liked having all my electronic tickets in the same place. When available, we reserved seats at a facing table for 2 or 4. Met some friendly travelers when sharing a table. Enjoy your travels, these are fun cities.

Lumo do NOT stop at York- from Edinburgh they are Morpeth, Newcastle and (some services) Stevenage.
While it is true that there are no Advance 1st class fares from Bath to York there are from Cheltenham Spa to York. So you book Bath to Cheltenham Spa, and Cheltenham Spa to York. Worst case you have to change seats at Cheltenham Spa.
It is still about 4x as much as Standard, but way more affordable. And the food is not too bad, way better than GWR anyway. Probably still not worth the extra money though.

A word of warning:> Bath to York (via Bristol) can be very expensive unless you split the ticket. It looks like 1 split at Cheltenham will significantly reduce the price. You stay on the same train at Cheltenham - sounds bonkers but true. If you were to purchase the ticket via the train company site (Cross Country Trains) , it will not tell you this and charge you the higher through ticket price from Bath to York.

I have just tested this via https://www.buytickets.scotrail.co.uk for Thursday 25 September leaving Bath at 10.46 for Bristol and then on to York to arrive at 15.30. The fare with the split at Cheltenham is £63.60 per person (without a Railcard discount). Scotrail do not charge a booking fee. On the Scotrail (or any other ticket retailing site), it is the 1 change option that goes via Bristol.

Sorry for asking. May i know what is LRH? You mean london Heathrow?

From what i know by using ChatGPT, from london to York takes about 2 hours. Same for Bath from London. From Bath to York, it takes about 4 hours.

Up to you.

may i ask if it is sufficient to spend only one day in York?

Thank you everyone for the helpful information!

LRH was a typo. Meant LHR for London Heathrow.

Brigitte, if possible you should spend two days in York. The free tour offered by locals is really good.