I visited London back in 2012 and took a train from Portsmouth to London and was dropped off someplace close to the London Eye. We crossed the river to check out the Parliament building and Westminster Abbey (one of my favorites from the trip). We walked up to Trafalgar to see that spot. It wasn't overly impressive but I'm not sure it tries to be since its a square. I am sure there are spots along the square which are great but we just scoped out the square itself. Then we headed through Pall Mall to check out what I guesss were embassies (super old but very fancy looking buildings) and then through Piccadilly. I'm not sure I got the point of Piccadilly, it all seemed like tourist shops. Scotland Yard was in the area as well as Buckingham Palace. We caught both sights before crossing Green Park which was absolutely gorgeous in September. Checked out the Wellington Arch at Hyde Park (another of my favorite spots). Then continued down to Harrod's on foot. Harrod's while monstrous is nothing amazing. Its essentially a department store.

From there, we took the tube to the London bridge mistaking it for the Tower bridge. The London Bridge is not what we expected so we walked down to the Tower bridge. Walked back along the north shore of the river, there was a nice little market area with quaint shops and pubs. We got lost, ended up walking around a dumpy part of town looking for some public transit to get back to a known spot but couldn't find any. Eventually made it to public transit, then headed back to a street with a lot of higher end shops (I don't remember what it was named). We had dinner there, hopped on the train and headed back to Portsmouth. We did all of that in one day, from about 10AM-8PM and mostly on foot. Overall, a good city but Edinburgh was the highlight of the visit to the UK. I'm not sure what the purpose of your visit is but if you can get to Edinburgh, take it!