England-21 days. Are these travel times realistic?

The Cotswold Motor Museum is at Bourton-on-the Water - which is quite a way from Bath. I think it would be better if you up sticks from Bath and spend 2 nights in the north Cotswolds (the Bourton-on-the-Water area). This also places you further north and hence nearer your next destination which I presume is Conwy.

Note that whether you drive from Bath or north Cotswolds, route planners will probably direct you to use the motorways (interstates) and north Wales Expressway (A55). You can pass through more scenic areas by using more westerly routes but the roads are not as good.

When you leave north Wales (Conwy), I suggest you spend a couple of nights somewhere near Chester (well worth seeing that historic city). You could commute by train into Liverpool.

Note that any hotel you plan to stay should have parking - especially in Bath & Chester where parking may be limited if in city centres.

Keswick to Durham via Hadrians Wall in 2 hours is a bit optimistic.

I would see Keswick to Haltwhistle (a nominal timing point for the wall) as 1 1/4 hours if you route via the A66/M6 and A69.
Then Haltwhistle to Durham as another 1 1/4 hours via the A69 and A1- planners probably give it closer to an hour, but once off the A69 there are some quite congested roads, and what seem to be eternal roadworks.

Personally knowing the area I don't think the M6 is the best route, just the fastest route. There are two much better routes in my opinion- the A686 over Hartside, and through Alston. That brings you in to the wall at Chesters Fort.

Or the B6412/B6413 up through the villages of the Eden Valley to Brampton. Brampton is well worth at least a brief stop for the fine stained glass at St Martin's Church if nothing else.

If taking the M6 route I suggest coming off at junction 42, not 43 as the Sat Nav will tell you. That takes you through the village of Wetheral- for the Priory Gatehouse, the Parish Church and the lovely walk to St Constantine's Cells (the sandstone caves above the River Eden).

From the wall to Durham the A69 is the most expeditious route, but certainly not the most scenic by a long way. But I'm not going to push that as I think you should have a night on the wall, but can't work out where to take it from, except maybe the Cotswolds. I'm not that fussed exactly where on the wall although I trend towards Corbridge, but then I'm a local, not a guidebook.

In the Cotswolds the Canals (the Stroudwater and the Thames and Severn, also the uppermost reaches of the River Thames) are in a different (more southerly and westerly) part of the District to really where most people on here think of. A little bit less of the chocolate box villages.
There is also the Oxford Canal on the eastern flank of the Cotswolds, but it's not really what most people go to Oxford for.
The upper reaches of the Thames, especially beyond the head of navigation, are undoubtedly lovely dreamy walks. Off hand I'm not sure how to do such linear walks by car, as I'm used to doing them from bus to bus either out of Oxford or Swindon.

York is amazing, far better than Liverpool. You need more time in York.

day 15 - 5 hours by train from York to London Kings Cross - is actually just under 2 hours or a couple of minutes over by train

York is amazing, far better than Liverpool. You need more time in York.

Well, everyone is different and just because one person loves a place doesn't mean everyone will. I enjoyed my time in York, but have no real desire to go back. However, I loved Liverpool and would go back in a heartbeat. It's a vibrant and fun city and there is a lot to see and do, especially if you're a Beatles fan but even if you're not.

While you're in Liverpool, don't miss the Philharmonic, which is a spectacular and very opulent pub. It contains amazing carvings, mahogany fireplaces, and mosaic bar counters. Don't miss seeing the men's loo—so gorgeous (have your husband sneak you in)! The Beatles used to frequent the Philharmonic many decades ago. In 2018, however, James Corden and Paul McCartney filmed a Carpool Karaoke episode and at the end, they wound up in the Philharmonic pub with Paul performing before a very stunned crowd (if you watch it, the pub scene starts at 15:30). The whole episode was really moving.

If you have time (and you may not), the charming cathedral city of nearby Chester is wonderful and worth a visit.

I think this itinerary makes sense taking into account the comments above about travel times etc. I like Conwy but for me it's a bit out of the way from the rest of your route and I wouldn't say it's worth it on its own to go so far from Liverpool. If by mountain hikes you mean walking in Snowdonia/climbing Mount Snowdon (which I definitely recommend- one of the easier major mountains in the UK to climb) then I would stay overnight in the area and also spend some time visiting Caernarfon Castle. It's quite an impressive one to visit and has a lot of history, even up to modern day as it's where Charles' investiture as Prince of Wales took place. If you have time you can even drive across to Anglesey which has Beaumaris castle.

Just some suggestions... good luck with the rest of your planning.