Spending a sunny, pleasant and blue skied day in London is easy. You'll just go to Hyde Park, drift further from there and stroll through Piccadilly Circus. However spending a cold, rainy and grey skied day in London is more probable and much harder so here are the things that I would do because to be honest, I would much rather explore London in the rain ( not like actually out in the rain don't worry ) than stay inside all day. For whats the point of being in another city if you're just going to do what you could do at home.
Drift in and out of coffee shops.
Take the tube to Old Street and find any coffee shop, carry a book or your laptop and slowly become a self proclaimed coffee expert within the 3 mile radius that you will explore. This is genuinely so much fun and the sound of rain, banana-walnut muffins, soft indie chill music and the smell of coffee create an almost perfect vibe.
Visit a museum, art gallery or an interesting exhibition at that time.
You don't want to waste a sunny day doing all of this so use the excuse of rain to explore the inside of London. There are a couple of places in London like Tate Modern, The Royal Academy Of Arts and Saatchi Gallery which usually have something new and interesting constantly going on.
Call a friend and create content together.
Find a place with a spectacular view like Sky Garden, St. Paul's Cathedral and Duck & Waffle and create whatever you like in the same space. I think I've mentioned this before but I think its magical when two people just sit together and create stuff, you don't have to be creating the same thing in fact they can be absolutely different and talking isn't necessary either. Whether its a research paper, blog posts, art assignment, robotics project, random sketches, a novel or even a scrapbook out of your pom pom collection. The actual content of what your creation doesn't matter, but the fact that you are together creating content in rainy London is what truly matters.
Find the cutest library you can ( some are Anthropology Library & Research Centre and Ziferblat) and find a random stranger to befriend. You never know maybe this person can accompany you on your adventures and even if its just a nice conversation, it'll be worth it.
Call a friend and go on a slow pub crawl.
Pick any bar in Soho and just go crazy. I feel like a huge part of London's culture is its pubs so instead of just focusing on the alcohol try and absorb the atmosphere, fish and chips and overall English vibes.
I hope this helped because I constantly see articles about how to spend sunny,warm and overall rare days in London instead of the usual rain or grey skies. I know that this so short so let me know if you want a longer version or even a part two!
Have a wonderful day,