Extended Visit Itinerary

Well, my trip was a bit different from yours, since I focused on England only (I spent 3 weeks last year in Scotland) but this was my itinerary, if that helps. I did rent a car for most of the trip, although I also took the train periodically.

London (5 nights)
Cornwall and Devon (9 nights - stayed in Falmouth and Plymouth)
Liverpool (3 nights - stayed with a friend)
The Peak District (stayed in Castleton for 5 nights) There is incredible hiking in this area. The Peak is an underdog when it comes to tourism and I really don't understand why, because it's so gorgeous. I really loved it here. In fact, I'd say The Peak and North Yorkshire were my 2 favorite places (and I've been to the Lake District and the Cotswolds and Wales before)
North Yorkshire (7 nights - stayed outside of Helmsley)
Northumberland (6 nights - first half in Seahouses on the coast; 2nd half in Haltwhistle near Hadrian's Wall)
Oxford (1 night)
Last night at Heathrow hotel

In the Lake District, if you're looking for uncrowded with good public transport links I would suggest 2 places on the edge of the district- Penrith and Cockermouth.
Cockermouth has it's Wordsworth history, and is a very nice town. And as of yesterday even has a new direct bus to Buttermere and Honister Pass for walking.

For a great garden get the bus to Workington, then the train to Ravenglass for Muncaster Castle). Go down there on a May or June weekend and we are getting a (free) bus link back to Muncaster (a 30 minute walk normallly), and to Wasdale for great walking- https://www.lakedistrict.gov.uk/visiting/getting-to-the-lake-district/wasdale-shuttlebus

There is a bus every 2 hours direct or every hour via Workington to Carlisle for the Cathedral and also the Tullie House Museum (which has an art collection). Also there is an art collection at The Beacon Museum in Whitehaven (with little known connections to L S Lowry, and also Matthias Read- quite a few of his paintings are in Whitehaven (somewhere I have a list of where all his surviving work are as part of the Visitor Information pack for St James' Church).

I can't immediately think where it is- but I am sure Cockermouth has a laundrette- there is certainly one in Keswick, one in Workington and (apparently as of last week) two in Whitehaven.

I've concentrated on Cockermouth, as I'm out west here, but Penrith would also be very suitable for the central and Eastern Lake district.

The Burrell Collection in Glasgow has reopened fairly recently after renovation and expansion. It was closed on my earlier trip to Scotland; I'm excited to be able to see it next year. Both Glasgow and Edinburgh have good art museums. Glasgow, while a large city, is relatively non-touristy--not an adjective one would apply to Edinburgh.

https://burrellcollection.com/

I managed to visit Yorkshire Sculpture Park in 2022 without a car. I was staying in Sheffield (which I don't particularly recommend); I think it would have been a bit quicker from Leeds. YSP is quite a large place and there's considerable walking required to get to the various sculptures; it's not a place to go if you only have an hour or so available. The sculptures are modern/contemporary.

https://ysp.org.uk/

I especially liked Bodnant Garden in North Wales. Although it was accessible via public transportation while I was staying in Chester, it took a lot of extra time to pull that off.

https://www.nationaltrust.org.uk/visit/wales/bodnant-garden.

There are a lot of stately homes and castles in Great Britain with lovely gardens. Accessibility without a car varies; the websites of the properties usually have good public-transit directions. After you have a rough idea of your itinerary sketched out, post it here and ask for good gardens in the vicinity of your planned stops. Although there are gardens so spectacular they're worth building an itinerary around, there are a lot of really good gardens that can be slotted conveniently into many Great Britain itineraries without adding a lot of travel time.