Late June with 2 teen boys: Bath, York, London

At York the Bar Convent- https://www.bar-convent.org.uk/ comes with very good reviews on this forum, super handy for the train station, and for walking into the city, and great prices.
For the trains get yourself a family and friends railcard (on line as a digital railcard or mailed to you) which gives you 30% off all fares (including the already cheap advance tickets) for the adults (the 16 yo is an adult) and 60% off for the 13 yo.
https://www.familyandfriends-railcard.co.uk/

PS- to go to York from Bath, book one advance ticket to London, then another from London to York- that works out way cheaper than a through fare. As you've been before you likely know that, but a handy reminder always helps.

I did not have the Bath to York train tip in my toolkit, and the reminder about the card was needed - thanks for that! That pays for a night's lodging, all told.

Will definitely check out the convent.

If you fly into Heathrow and want to go directly to Bath, the National Express coach service is an option. https://www.nationalexpress.com/en/destinations/bath/heathrow-to-bath Get on the bus at Heathrow, settle down in a comfy seat and get off at Bath. I've heard those over-the-road coaches (aka buses to us Americans) have comfortable seats and free wi-fi. I would take that coach in a heartbeat vs taking a train in to Paddington and then another train to Bath. Cheaper and easier. EDITED to add: I can't tell if you want to do some sightseeing between landing in London and traveling on to Bath. If you do, I'm all Emily Litella: never mind. :-)

Just consider that getting from Bath to York is going to be one whole day of travelling, and you only have a week. Is there somewhere else you’d like to see, or maybe just choose 2 places, like York and London or add Durham which is near York.

Absolutely the coach to Bath. It's cheap, convenient and direct. Just sit back and relax after your TA flight.
The only similarity to Greyhound is that they are coaches.
Far preferable to rail routes via Paddington (or as I saw the other day someone suggesting local trains to Reading avoiding London and changing there, to save literally 10 pence) or even the illogical rail coach link to Reading, changing to coach there.
Book ahead and the four of you should get that for as little as £38, or not much more.

Suz, Emily Litella! Forgot all about her, Gilda was the best. Enjoy the wit and humor on this forum almost as much as the travel tips! Thanks for link and mention of coach to Bath, we sometimes forget bus is easier and more direct.

I was able to book at the Bar Convent - this will be a feature. THANKS for the referral!

Family schedule changed, so lost my weekday reservation at 3 Abbey Green/now Eight in Bath, couldn't find something there for the weekend we arrive - found London chain hotel instead for first nights, then to York on a Sunday train, then all the heck the way back to Bath and another B&B with triple family room, and then Friday back to Heathrow.

NEXT trip I may go alone...'

Thanks for the help!

I wouldn't dismiss the Railair coach to Reading as "illogical" compared with back tracking to London although it usually makes more sense (pricewise) if being used as part of a return rather than one way. I would put the direct coach to Bath clearly ahead these days as the frequency is better today than it was. Avoid the ones that require a change at Bristol though.

In Bath, check out the Brooks Guest House and nearby Scallop Shell Restaurant.

In York, The Minster Hotel

Both at great locations.

I was able to book at the Bar Convent

The Bar Convent is great fun. Don't miss the wonderful chapel, hidden away under the roof when it was banned to practice the Roman Cathoilic religion in England.

From the OP's latest post:
"back to Bath and another B&B with triple family room, and then Friday back to Heathrow."

The coach (bus) will be ideal for this journey. It will take you from Bath train and coach station to Heathrow coach stations.
Once again, the coach is the best way to go.
We have taken the coach both ways (to and from) Bath (coming from Heathrow & going back to Heathrow) and were pleased with the ease of the journey, comfy seats, clean coaches, great WiFi onboard. And price.