Hub2dee
When you say "easy reach of city attractions" do you mean driving or on foot? Parts of La Jolla definitely fit your bill, if you can afford it.
Anyway, If I had a winter to spend there I'd get one of the holiday lets (rentals), Bayside, in South Mission Beach. [My dad lives on Bayside Walk, so if you end up going out there, I might even introduce you....] It's quiet in the winter (October-Easter), I think it's still affordable in the winter (they mostly rent to students; the rental rates are astronomical in summer). There are no cheap food shops in walking distance, but you can just stock up when you do go shopping. The beach might literally be your front garden if you can stretch the budget to afford Ocean/Baywalk frontage. It's so nice there you wouldn't mind the hike to shops or anything else.
San Diego is divided by a poxy river (SD river) that occasionally has major floods & wipes out most of the flood plain housing & shops (yet they still built on it...). River goes down Mission Valley. It used to empty out where the airport is, but was channelled in the 1940s to go out between Mission Beach & Ocean Beach.
Both sides of the valley there are big mesas (plateaus rising above the valley floor). North side is (west to east) Kearny Mesa to San Carlos, up to Mission Gorge (go east of the Gorge to get to El Cajon). South of Mission Valley is another mesa which is the more established neighbourhoods of SD. I grew up on the south mesa: Mission Hills, Golden Hills.
The south mesa slopes off to down town, the harbour, the Barrio, South East SD (ghetto....)
Most of these places (even SE SD) traditionally had their own very strong community feel, and to me they are the heart of SD as I know. It's a very spread out "heart", but still places with firm self-identities.
Downtown is funny it did have its own community of very poor homeless transient types, I'm afraid to say. But it's increasingly trendy to live there, now, not least to beat the congestion. A new baseball stadium is being built, so the trendiness will only increase.
The thing is, if you want "some nature" but still close to services... not much of San Diego proper is still like that. Because I don't live there it's hard for me to be sure, but I suppose Encinitas & Oceanside would be your best bets. Because of the hills & mountains, though, you can go 5-10 miles inland & still get an Ocean view. It's the "natural space" that is very hard to get. I suppose some of the newer developments in south bay area (south of Logan Heights, north of CHula Vista) might suit, but I don't know much about them. Many of San Diego's canyons in established areas (HIllcrest to Kensington) can't realistically be built in (too steep, prone to erosion, important as urban wildlife corridors, too likely to undermine existing buildings), so this is where you might find some perception of natural space.
But coming from Britain, you might not perceive the natural vegetation (mostly chapparal scrub) as "natural space", it just looks like scrubby dense undergrowth. Green meadows and woodland are unnatural vegetation in this part of the world; if you want green lawns & trees, you may as well stay in the city, esp. near Balboa Park.
Can't believe I wrote so much! Hopefully I know what I'm talking about. If you really want natural space, you might want to consider some of the communities further north: Laguna Beach, Oxnard, Santa Barbara. Carmel (as you said).
DD is clamouring to play cbeebies. hth