England, Wales & Scotland in 3 weeks? Doable via rental car?

Google is a good starting point, particularly their Maps site.

We did England and Wales for four weeks this October.
Rented a car from Heathrow, returned at Southampton.

Itinerary was Bath, Glastonbury, Wells, Tintern Abbey, Cardiff, several castles in S. Wales, Tenby, Wales, St. David, Wales, another village in S.Wales, Chipping Campden ( Oxford, Blenheim Palace, Stratford upon Avon, the Cotswolds), Warwick, York, Castle Howard, Pickering, Yorkshire Moors, Whitby, Durham, Hadrian's Wall, The Lake District, Winchester, Southampton (cruise home to USA).

Make sure your rental car has a navigation system or you have an up to date UK GPS.

If you want to see the countryside well, a rental car is perfect, but you must consider where you will park.

If you want to see where Doc Martin was filmed, you'll be driving out to Cornwall, one of the more remote parts of Great Britain. For details, see https://www.visitcornwall.com/about-cornwall/blogging-cornwall/doc-martin-locations-to-visit

There are plenty of ruins, antiquities, scones, etc. all over England, Wales, and Scotland so you can go just about anywhere and not be disappointed on that score. Invest in a good guidebook like the Rick Steves one (I'm not affiliated with RS in any way). The cost of the guidebook will be repaid many times over in savings when you become an informed traveler, knowledgeable about where to go, where to stay, where to eat, etc. that is a good value.

As for watching fabric be made on site, you may have to go all the way to the isles of Lewis and Harris to see Harris tweed. The Lace Guild site may give you clues as to where lace is made that visitors can observe https://www.laceguild.org/ Others on this site may have other specialized suggestions if you start a thread (pardon the pun, LOL) dedicated to that topic.

Doc Martin exteriors are filmed in/around Port Isaac on the northern coast of Cornwall. I have no idea whether they also film at a studio elsewhere. I didn't go to Port Isaac myself but did travel around Cornwall. I especially enjoyed the many gardens I viewed. And there's definitely no problem finding scones in Cornwall.

I spent six weeks in England and Wales this year but didn't get as far north as York, so the prospect of adding Scotland and trying to cover that amount of ground in three weeks concerns me. Certainly you can do it, but you'll have a much higher percentage of time on the road than I would like. I'd recommend scaling back the extent of the trip and planning to return.

The Cornish town of St. Ives is known for its art galleries; you'll find ceramics there as well as paintings. I spent several days in St. Ives, enjoying the shops and just walking around. Most of the shops are fairly near the water, but the David Leach Pottery is more than a mile inland and distinctly uphill.

London has always been a great destination for crafts. You could literally spend days in the Victoria and Albert Museum alone. I was in the city for ten days and didn't manage to see all of the V&A. You should prioritize the types of work you want to see, else you may not have time for much else in London. The V&A shop is very good.

Here are a few craft galleries I enjoyed in London:

Vessel Gallery: Primarily glass; magnificent and $$$$. Think of it as a museum with price tags. Coincidentally, located near Portobello Road Market.

Contemporary Applied Arts: Glass, ceramics, fiber, jewelry, etc. On the South Bank, a reasonable walk from Borough Market, the Tate Modern, Shakespeare's Globe and (a bit farther) the National Theatre.

London Glassblowing: Glass, some of which is blown on site; observers welcome. Also within walking distance of Borough Market. LG is farther east than CAA, but you can combine visits to the two rather easily if you don't mind walking.

This article from Architectural Digest may be of interest.

The Crafts Council has a directory up on its website. You can search by location. Members tend to be high-quality crafts people. Keep your eyes open for a free brochure listing galleries any time you're in a top-end craft gallery. Something like that is often the easiest way to find other spots of interest in the area. Also ask any tourist office you come across; unfortunately, there aren't a lot of those around because of budget cuts.

In Oxford I enjoyed the Pitt-Rivers Museum, whose ethnographic collection has a lot of what I would call "crafts" from Asia, Africa, the Americas and Oceania.

In Bath, the Museum of East Asian Art is lovely and has many lovely handcrafted items.

Norwich has a sort of combination shop/museum that focuses on south Asia (the affordable [South Asia Gallery][7]) plus a fine museum of Asian art and decorative art (Sainsbury Centre for Visual Arts).

There are obviously many, many more possibilities. Google "craft galleries" and the name of the city or town you're interested in.

Another useful planning site is National Trust UK. It has an extensive interactive site which covers all their properties and parklands in England and Wales that they maintain thru membership and volunteers. It is well worth becoming a member (you may be able to join on a couples pass) as this will cover admissions and parking at hundreds of locations. They did have an Overseas Visitor pass; it may still exist if you search their site. English Heritage also has a site. I think some of their Properties may overlap with the NT UK, but there may be others mostly archaeological sites that may be of interest. Scotland Also has their own National Trust. Visit Scotland has an excellent interactive site and they also have a pass available. Historic Scotland has a list of all the castles. Secret Scotland is supposed to take you off the tourist trail. We have gone on British road trips twice and were able to travel to all 3 countries. One was for 21 day in September/ October. I would start in the North if you go then. The second trip was for 15 days in April/May...we still had snow flurries in Scotland. The big advantage of not going in the Summer is that you can go on the fly without reservations unless you hit a Bank Holiday. Dress as you would in Seattle. The British are outdoorsy. Just make sure you wear wet weather shoes or boots. Wear fleece and a raincoat layer. Wear a knit hat as it can be very windy. We spent a almost a week each time in London: it's great for antiquities, history, shopping and walking around town. Wales is best for ruined castles. Scotland is best for whisky tours.

You could spend 3 weeks in each, and not see it all. Is it doable, yes, depending on how selective you are choosing which sites to see. Don’t spend so much time getting to places that to don’t have enough time to see and enjoy them properly. Down in Cornwall, where Port Isaac (Doc Martin location) is filmed the roads wind a lot and are narrow, so it takes longer to get to where you want to go. Some episodes of Doc Martin have also been filmed in Truro and castle hotel in Tintagel. You may enjoy the area in Scotland with the woolen mills and stores near Peebles, Hawick, and Elgin (The borders region). Near there is Jedburgh abbey and Melrose abbey and some castles too.

Do not expect to rely on having a data connection for navigation for cell phone- data, and cell service in general, can be spotty in rural areas. Go with either a phone navigation app that still works while offline or a standalone GPS.

English Heritage also has a site. I think some of their Properties may overlap with the NT UK,

There are no properties that overlap between English Heritage and the National Trust. We have membership for both as we found it impossible to choose between the two however I would suggest looking at which organisation maintains what and decide which one offers more of what you want to visit.

To answer your post’s question, yes it is doable. You just have to decide how much you want to see, and what can be put off until a later trip.
We visited England, Wales and Scotland in 26 days this past summer. We started with Rick’s suggested itinerary for 3 weeks, and adjusted for our interests. Cornwall would be a reach, but maybe doable if you kept to southern Scotland—still lots to see there.
We did 5 nights in London; 3 in Bath, with trips to Stonehenge, Avebury and the Cotswolds, then picked up a car; 3 nights in Conwy, Wales; 2 nights in York (which involved zigzagging a bit more than necessary but we wanted to visit York ); 2 in the Lake District ; then 9 nights in Scotland. We went to Oban, Inverness, and Edinburgh for 3 nights each. You could reduce the number of nights in Scotland to fit your interests. I know I have not addressed your stated interests, but you can do the 3 countries if you are willing to make some compromises.

There are no properties that overlap between English Heritage and the National Trust.

There are some, but it is fairly limited, usually where by a twist of history each own or manage part of a greater whole, so membership of one gets you into everything. One example is Stonehenge where EH own the stones themselves but NT a lot of the surrounding landscape and Hardwick, where EH are responsible for the Old Hall and NT the New Hall.

Don't lock yourself into roundtrip flights from Seattle to London before your plan your itinerary. While these may be the best flights, you may do better starting or ending elsewhere in the UK. With a change in London or Amsterdam, you can get to all kinds of airports; for instance, you may want to start in Scotland and work your way down to London, without making a loop.

You can also fly out of Vancouver BC to London. You can take an airport Shuttle; there are a number of pickups near the I-5.