I'm from LA originally, and I would attract odd looks for wearing a sweater in the summer.
As for being cold indoors, it really depends on the house, how well sealed it is, how good the heating is, insulation, etc. My best friend lives in the exact style house as me, built in 1899, just a few doors down. Our house is usually very warm and I find hers cold in the winter. Oldies also tend to have their houses colder as heating is pretty expensive here, I think.
The upstairs of our house is quite cold in the winter, but we have an electric underblanket which my husband tolerates because he's lovely LOL I switch that on for 30 minutes or so just to warm the bed through and then I'm fine all night.
You won't need thermals for day to day use, but they might be an idea if you spend long periods out doors in Jan/Feb.
Remember, it's much colder out of town and up north than it is in London.
As Aless says, it's the damp you need to worry about, especially coming from the desert. It gets into your bones and once you get damp-cold, it's hard to get warm.
If it's cold and frosty, then it's dry and it's easier to stay warm. If it's drizzly or misty or damp, you get cold without it actually being that cold outside.
I find that warm socks, boots, jeans, long sleeves and my North Face down jacket are enough for me for outdoors.
Indoors, if it's chilly, 2 thin layers on top, tracksuit bottoms and socks keeps me pretty warm.