I do think not having a car would make some of these very very slow to get around. If you have a car it would be quite easy. It isn't intimidating to drive in England as long as you avoid London and pick up your car in another city and/or ditch it prior to going to London. I've done this 3x and have basically driven around the country for a total of 6 weeks. Roads are generally well maintained, drivers more polite, and you get used to the opposite side quite quickly. Cornwall is rather "colorful" (extremely narrow lanes) but still a feasible place to drive if you are careful. I'd take driving there any day over my normal Los Angeles crazy :0! yes it would increase your cost for the trip, but it would allow for much more flexibility in seeing places.
That being said below are my suggestions from the places I've visited:
London - I would spend 6 days in London, then 2 more for your "day trips" . Brighton is just "ok" not really what I expected. The royal pavillion looks lovely from the outside but inside there isn't much left except replicated things since all of the furniture including most of the rugs is in houses that the royal family has. I found it really dull tbh.
With the remaining 27 days devoted to:
Liverpool
Chester
Hadrians Wall- I did this in like a day (but I had a car)
York - 3 days is good.
Lake District - 4-5 days- The Lake District is AMAZING! I'd recommend staying near Windemere/Ambleside . Personally the Lake District to me was the best place in all of England.
Cambridge
Oxford- 3 days should be ok.
Cotswolds - (quite dull to be honest) 3 days MAX. They have a really intriguing wild animal park you should check out- Rhinos and all.
Eden Project - This is way too far south I would cut out Cornwall entirely OR Fly there first if you are able ???
Seven Sisters
Lands End- I found this very dreadful (it is a mini tourist trap) and St.Ives is about 1,000x more interesting- they have some great arts and crafts shops that are authentic in St.Ives. St Michael's mount in Cornwall is quite nice if you really insist on doing Cornwall. Penzance is small but has great restaurants.
Black pool
Portmeirion - Wales
Snowdonia - Wales - lovely, massively large and spread out. 1 or 2 days depending what you enjoy doing. Skip "fairy glen" it is overrated--- my friend and I took 2 hours trying to not slip on moss to get to it. They have some very interesting neolithic sites that are kind of in the middle of nowhere and may require some good "duck boots" due to cow droppings/needing to cross fields.
Brecon Beacons - Wales- 1-2 days
Hay-On-Wye - Wales
Conwy - Wales - CONWY IS THE BEST! that being said it is super small and you can see it in 1 day or half a day. My friend and I (although we did have a car) did Conwy and Carnaferon in a day along with Beaumaris. I think Canafareon (yeah I'm probably spelling this oddly) was the best of all 3 castles, with Beaumaris coming in 2nd.
Llandudno - Wales . - lovely seaside area. I used this as a base when in wales. Don't miss Bondant gardens!!!! Which is not far away from Llandudno. 2 days?