(I'm from Utah. Obviously not a Brit.)
We went with my two oldest kids in June. We stayed at the Holiday Inn Express in Hammersmith. It was about 1/4 mile walk to the closest underground station. It seems like we paid around $100/night, and it had Wi-Fi and a continental breakfast.
I bought a 3 Sim card for my unlocked Motorola smartphone. There are 3 stores everywhere. I bought it in Edinburgh and it worked everywhere in the UK, as expected. Data plans were much cheaper there. I'm sure I had some minutes and texts on the plan, but I didn't use them.
You're not planning to drive in London, are you? Don't do it. The traffic, left-side driving, and congestion fees will kill you. Get an Oyster card, load it up and take the underground everywhere. It works great.
We arrived from Edinburgh at St. Pancras/King's Cross. No problems there, but we weren't there at night.
Stop by Grosvenor Square. It's right next to the US Embassy. There are two really impressive statues there.
I also went to the showroom of Holland & Holland (as in 375 H&H Magnum). They had double rifles and shotguns in there that cost way more than I make in a year. I did make one purchase. A baseball cap with the H&H logo. Only thing in the store that I could afford.
I didn't bring any knives with me at all.
Felt naked the whole time.