On the left side of Buckingham Palace, there is an art gallery within the palace. We saw a DaVinci exhibit when we were there in 2019. Up the block from that is the Royal Mews where you can see the carriages that have been used by the royals, along with some of their horses, throughout the years. My favorite tea store is across the street, Whittard of Chelsea is from where I order all of my tea. SO good! It's not a cafe, just a tea store.
Stroll through St. James Park. It's beautiful and across the street from the palace. At the end of that, you will come upon Trafalgar Square and the National Gallery. Remember, all of the museums in London are free of charge to visit, unlike in most other large cities.
With your back to the museum, you can walk down to the Banqueting Hall, through which, when Whitehall Palace was still there, King Charles I was escorted through it, had to climb out a window to his execution to the dais that was raised for the occasion. It's the only existing building from Whitehall Palace.
Across from that, you can glance down the way to see the Prime Minister's residence at 10 Downing Street. I remember standing in front of the door when I was 20 and visiting, but terrorism laid an end to that.
Continuing down the main road, you will come upon Parliament and Westminster Abbey. You can also stroll along the Thames, if you wish, or cross the bridge and take a ride on the London Eye. Oops, I almost forgot... the Churchill War Rooms are also just before you reach the Abbey, on the same side as 10 Downing Street.
If you want to learn more about the Tower of London, search for Beefy 409 on Facebook. He is a Yeoman Warder at the Tower and gives tours via his web page. He also talks a lot about the ravens at the tower and the squirrels. He had one squirrel who would jump onto him for a peanut, but sadly, Cyril has passed away. It's nice history to know before you visit in person.