What a great opportunity way too few Americans get to experience. You’ve already been given some great advice. As an American who lived in west London (Chiswick) for a few years before moving to the continent, allow me to add on.
For location, given that it seems like your girlfriend will be working somewhere in London, some things to consider. Transportation was mentioned. Depending on where she is working, I’d be sure to be near a tube stop (Chiswick and Richmond are good candidates on the west side, see reference re access to Richmond Park above and Surrey, including Box Hill etc, I’ll return to this) or a rail station if her office is near a train station in the City. I’d suggest accepting that you’ll want easy access to London for the sights, theater, food (there’s a good Chinatown), museums, etc. Depending on where you are, she could also cycle to the office. There’s also the option of living farther away and taking the train in, this takes me to the next point, food. ** Just saw your girlfriend’s office will be close to King’s Cross, so she could plan on taking a train into the City, but read on for considerations. Do pay attention to transfers getting into / out of the King’s Cross (or which stop she ends up using), at least be aware.
You could get out and into small villages, but an advantage of say Richmond and Chiswick is the variety of restaurants for dine-in, take-away, or delivery, a variety you won’t have in a smaller village. Regardless, you’re likely to have a good Sainsbury (think Ralphs) or Waitrose (think Pavilions, not quite Gelson’s… considering you’re coming from CA these should resonate) to walk to, which is also a consideration for a location. I’d pick a place you can easy walk to the market, including butcher and bakery (and pub and cafe), which won’t be that hard. Also, know that you’ll be shopping for groceries a good three times a week because things are fresh and without the preservatives in US foods. I “joke” there’s no need for a large refrigerator in a UK home because, well, the food doesn’t keep as long. No Costco runs… Honestly, I’d caution against moving to a quiet village as your first stop because of the increased need to drive, variety of services, etc. Your mileage will vary, but considering the desire to enjoy, there are great options to be outside the City while also enjoying easy access to it and other access afforded by access to the City as a hub.
This segues to why we picked Chiswick, which may or may not be helpful to you. At the time my wife was working nearly at Oxford (ie way outside of London) and had to drive, and this put us outside the “bubble” of London track (or at least on the edge) with the tube (District line) easily taking us into London or Heathrow. We were very close to Turnham Green tube, had a large park steps away, great options for food and entertainment (Westfield mall at White City for movies), easy for cycling and running, and yet it was quiet.
To hype Richmond, you have access to the Thames and the tow path, the park (which for the early riser is closed to cars until I forget when), and rides south. You can easily pedal far afield or hop the train with your bike to get farther afield. You also have great outdoor areas along the Thames for lounging and eating during the warmer months. From Chiswick, it’s an easy pedal or walk to pubs and restaurants along the Thames in both directions. (I preferred living in Chiswick than Richmond which I visited often, FWIW.)
Hopefully, you’ll be able to do a visit to get a feel for whatever place you want to live in, including food options, pub (great beer, great food, great relaxation… if it’s the right place), park options, access to cycling, access to London or elsewhere. An advantage of being near London is access to Eurostar and training it to the continent. Pop over to Paris (or Disneyland Paris!) or elsewhere. Or take a train to the North or the West, hop over to Ireland, or a just as easy flight to Spain, Majorca, Italy, Switzerland, etc.
Saw your comment about retiring somewhere… I’d suggest looking at Spain as well. 
Sorry for the ramble, I could probably clean it up, but trying to get this to you before getting out. Happy to share more here or PM.
) She’s really into that lifestyle too. Hoping we can adapt!
As long as we can choose to be there it’s good enough. Just like SF. I don’t go there for months (though just 20 mins away) but nice if you wanna visit for a couple hours for night out / lunch / cultural activity etc
