Not going to address Paris ( been there 3 times) but will share intel on London. Its my favorite city on the planet.
Predicting weather is a lost cause but a waterproof rain coat is a must.
London and Paris are very popular tourist destinations. Not only for yanks but for other Europeans, travelers from Asia, South America, Canada, etc. Won’t be as crowded in March as say June or July but won’t be dead either.
Yes I would use public transport ( Tube and trains, maybe a bus or 2 ) to see places outside London proper. Easy 1 hour train ride from London to Cambridge. Windsor Castle is about equal distance.
Rye is a pleasant 1.5 hour train journey so is the coastal town of Whitsable.
If weather is good the massive Richmond Park is wonderful to walk about. Hopefully you’ll get to see the fallow red deer.
Walking the Thames River path is enjoyable.
There is so much to see and do in London:
Theatre, free museums, street markets, galleries, music venues are just some of the things of interest.
Been visiting for decades. Never fail to find something new to explore and enjoy. This past July it was the Shoreditch neighborhood murals. In November it was Bletchley Park.
Pack light, dress in layers, wear well worn comfortable shoes, take gloves and review such websites as TimeOut London, Visit London, EaterLondon, and https://officiallondontheatre.com/ websites a couple of months before you go.
Pubs are often great spots for lunch, resting tired feet, observing a dart game, a game of pool, etc.
Pub lunches with bowls of soup and stews often hit the spot to ward off the chill. Look for sidewalk signboards advertising their lunch offerings.
The Dove in Hammersmith is a favorite of mine after I enjoy a walk along the Thames path. Same with The White Cross near Richmond.
Be forewarned though not all pubs serve meals. The Jolly Gardners ( 36 Lower Richmond Road ) doesn’t but great proprietor and nice pint of Guinness.
Coffee shops are rampant now. The evil empire of Starbucks everywhere. Thankfully better options: Costa, Cafe Neros, Harris and Hoole….and a great new find for me in July, OAT coffee in Bricklane.