England walking base, no car

I will suggest you look at HFHolidays. This is a member-owned cooperative ( not a tour company) with large country houses all around the UK in great areas for walking. They offer 3, 4, and 7- night programs with guided walks on 3 different levels, plus room and board, transport to the trailheads, and easy instructions on how to reach the location by public transport. Basically all you need do is show up, and decide which walk you will do each day.

https://www.hfholidays.co.uk/holidays/guided-walking/guided-walking-holidays

They also offer walks Mon the long-distance trails from bases in their country houses, with transport to and from the start and finish each day. And week-long progrmans in Europevand around the world.

We have stayed and walked with them from their locations in the Lake District and Jurassic Coast, and will be joining them in the peak District in April. On our list for future trips are their houses in Cornwall, Wales, Scotland, Exmoor, and Yorkshire.

Cotswold Way.

https://explorethecotswolds.com/walking-the-cotswold-way/

Too many choices but have a look here https://www.theguardian.com/travel/2021/jul/31/six-best-british-uk-waking-walkers-towns-hubs

I'd also suggest Keswick in the Lakes, Buxton in the Peak District and Skipton the Yorkshire Dales

If you loved Cornwall you could go back there.

Hebden Bridge in the article is a really good idea, an area which sadly seems to be very little known to members of this forum.

In the Lake District there are lots of places beside the ubiquitous Keswick that you could stay in, but are unknown on here. 4 miles east of Keswick you have the village of Threlkeld (on the same X4/X5 bus route for connections to rail at Penrith), and 2 miles west of Keswick there is the village of Braithwaite.
Both lovely little villages, in very easy reach of Keswick.
Penrith itself, on the main train line from London to Glasgow is a great place to stay, for walking in the comparatively little known Eden Valley, as well as walks in the Ullswater area or in the area 'back of Blencathra' (between Penrith and Threlkeld). The Eden Valley bus service is about to improve markedly from the end of March with new bus links to Kirkby Stephen- timetable awaited. When that comes out it may make Kirkby Stephen viable as a walking base without a car for the first time in over 20 years.
But do look seriously at Lola's HF link- especially the house at Sedbergh- the Howgills (where Sedbergh is) has wonderful, but almost unknown walking- just as good as the Lake District but without all the crowds there. It is also on the new bus route mentioned above.

Around England and Wales there are literally hundreds of possible suggestions.

Here is the direct link to the Sedbergh house in the Yorkshire Dales, mentioned above. Use the “photos” button at upper right to see the scenery on offer—- rivers, bridges, waterfalls, and more. And open the “accommodations” in the menu par to see photos of the country house itself.

https://www.hfholidays.co.uk/holidays-and-tours/3-night-western-yorkshire-dales-guided-walking-holiday

Yjisbis one on many multi-day holidays offered at this and other locations by this group.

My vote would be for the Yorkshire Dales; maybe Grassington as a base? Here is a recent thread of mine about a base in Yorkshire that might give you some ideas about it. I will be spending a week in Yorkshire in mid-April. https://community.ricksteves.com/travel-forum/england/moving-on-to-yorkshire

Another suggestion would be The Peak District, which is known for its lovely scenery and available hiking. Here's a another thread of mine that gives more details: https://community.ricksteves.com/travel-forum/england/next-stop-the-peak-district

And if you'd like to stay in a place with access to antother coastal walk, there is Northumberland, where you can travel the gorgeous Northumberland coast and enjoy the beautiful and dramatic scenery here. https://northumberlandcoastpath.org/the-route/route-map/

Thank you everyone for the great suggestions.

You’ve gotten some excellent suggestions. I especially enjoyed Yorkshire Dales (stayed in Grassington and Richmond), the Lake District (stayed in Keswick), and the Peak District (stayed in Bakewell). Other great options are SW Wales (stayed in St, David’s; the coastal walks are breathtaking, and the buses along the coast are ideal for hikers; this was pre-Covid) North Yorkshire Moors (stayed in Whitby; you can walk from there or take a train to other hiking spots), and Hadrian’s Wall (stayed in Haltwhistle; we booked a guide, but I believe you catch a bus to good starting places for hiking; we also walked to the wall from our B&B).

We only wish we had discovered HF Holidays years earlier. We have done five of their UK houses… Wales, Scotland, South Downs, Jurassic coast and Cotswolds. Each house is unique, each location has so much to offer and our groups have been friendly and welcoming. Usually we are the only Americans which we also find to be so much fun. We will be doing a rail trip starting in London to the Black Forest with HF in June. I’m sure it will be just as fabulous.

I'd also suggest HF. Friends of ours took a few days at their Bourton-on-the-Water location in the Cotswolds, and they liked the walks and the house very much. It is right in the centre of Bourton-on-the-Water but away from the tourist route. There are short and medium one ways and out-and-backs all around.

While Bourton-on-the-Water does not have a station it is served by buses from Cheltenham and Moreton-in-the-Marsh, both of which do have stations.

The Cotswold Way passes nearby.

If you want to do everything yourself you could base in one of a dozen villages and small towns - the countryside is excellent for walking. (a bit muddy at the moment since Noah only just left with his Ark)

There's a route called the Monarch's Way, which stretches from Worcester to Shoreham-by-Sea, and roughly traces the route of King Charles II's escape from the Battle of Worcester being pursued by some of Cromwell's lot. It's a funny route in that it twists and turns to reflect the various times Charles had to switch his plans for various reasons, such as being cut off by hostiles, or as simple as a broken bridge. Allegedly he once had to hide inside of an oak tree, which is still preserved along the path.
I've done bits of the walk, namely the branch from Crewkerne. It is easy to follow and strenuous in parts, but because Charles generally avoided the larger cities, it passes through mainly countryside towns and villages as well as some historic places.

Wikipedia gives a very riveting and detailed account of Charles' escape route. It may well be one of the most entertaining Wikipedia entries I've read. It's definitely worth a go if you've got some time. The full route takes about six weeks to complete but you can easily start and stop in many places.

Here's the route, as regaled by legend on Wikipedia:
https://en.wikipedia.org/wiki/Escape_of_Charles_II

and here's the route, as told equally as well by Shropshire's Great Outdoors:
http://www.shropshiresgreatoutdoors.co.uk/route/monarchs-way/

I walked a bit of the Monarch's Way from Charmouth! Lovely. I forgot about this one. Thank you.