Was in London in January of 2019. Stayed in Ealing at a friends flat near Ealing Studios. Weather was cold but not chilling. Mostly blue skies but a couple of days had early AM drizzle. Dressed in layers.
This is my second reply to the OP’s post.
Saw Hamilton, (trained into and out of London on TFL.) visited farmers markets, walked along the Thames Path.
Met up with Nigel and Carol and they took me to see Peterborough Cathedral as well as Eleanor Crosses.
Discovered the Booking Office restaurant and bar at St Pancras. Great find. Had tea with a dear friend there. Also discovered Foster Books in Chiswick.
Another fun find was the teeny Dr Expresso Caffetteria ( their spelling not mine) adjacent to the River Cafe. Up and around the corner Hurlingham Books. All three I would return to in a nano second. All by the Putney Bridge Station. It was from this area I began my stroll along the Thames Path to Hammersmith Bridge. Lots of dog walkers, parents with strollers, rowers on the Thames and meanderers like. Blue skies that day.
My point being weather in January that year wasn’t freezing or very rainy. Might not be the case in 2022 but no one can truly predict what mood Mother Nature will ever be in.
So many great museums, theatre matinees, shopping arcades, restaurants, cafes, pubs….never a problem to get out of the cold or wet weather.
As far as neighborhoods to stay Ealing isn’t bad.
15 minute ride on a TFL train ( covered by the Oyster card) from Ealing Broadway to Paddington Station.
There’s a Premiere Inn in Ealing. Also a Travelodge and IsBis hotel. Haven’t stayed in any of them so reading reviews on TripAdvisor would be wise.
Good Vietnamese food at Pho Saigon, Coffee and scones at Ginger and Moore. There’s a Wagamama, Bill’s and Nandos. Chain restaurants but all 3 quite good.
Pretty certain a number of rentals via AirBnB. In doing your research be sure to find out how close or far they are from the Ealing Broadway station.
Or see what rate you could get at the Westminster Premiere Inn Hub. It’s right in the heart of London. Would save you travel time as it’s within walking distance of Parliament Square, Churchill War Rooms, The Thames, Trafalgar Square, St James Park, Buckingham Palace. Small rooms but practical and a good price.
Yes it will start to get dark around 3:30-4pm but that means nothing. You’ll find lots of people out and about along sidewalks.
In planning your days know that most sites open at 10am and many close at 5pm. For instance Tower of London January hours are 9am-4:30pm. Make a list of what you must see, research hours, and plan each day accordingly. Many of the museums have no admission fee. Special exhibits will have a fee though.
Look for sandwich boards outside pubs, cafes and restaurants, that will list lunch specials, buy pre made sandwiches, or meal deals at Grocery stores like Tesco and Sainsbury… look for buffet meals at ethnic restaurants…
For same day theatre tickets check the Half Price Ticket booth in Leicester Square.
Get an Oyster card ( refundable deposit). Put 15 pounds on it . The Oyster provides access to busses, the underground tube trains. Depending on how often you use it you’ll probably have to top up but that’s so easy. All tube stations have machines.
Use cash but carry 2 credit cards. Get cash from a bank ATM. If you have one use a Capital One card. It’s the only CC that doesn’t charge a fee for cash withdrawals…although double check that is still true.
Yes London is expensive but with research and careful planning you can afford a week long visit without going broke.
Great city. Enjoy it.