We spent time in London in May of this year, so I can confirm that you can buy the Oyster card at the tube or the Heathrow Express/Connection station using a Chip & Signature cards. It will not ask for signature. I was able to get a refund at the tube station with no problems. I was told I could get a refund at the counter too. We had no issues with our American credit cards.
The Heathrow Connect is cheaper than the express and takes just 30 mins instead of 15. Good if you want to go to Paddington. Other options may be better for other destinations.
We found our smartphones indispensible. You can buy a SIM card there. Here's a discussion
http://www.early-retirement.org/forums/f46/best-data-network-in-the-uk-86348.html
I didn't see a Parliament tour mentioned. It is amazing and our guide was fantastic! In spring, tours are probably only on Saturdays. Book ahead. I recommend the guided tour. It was the one of the highlights of this trip to London.
We visited Kew Gardens for the first time and I think it is a great middle of your trip destination (if the weather holds). Alternatively, you could schedule a day at Hyde Park/Kensington or a trip to Greenwich & straddle the Prime Meridian. Greenwich is a cute town too.
I found Rick Steve's Westminster Walk to be good, but the other walks weren't as interesting & some, like the city walk, are long.
London has great restaurants and pubs. Pub food is much improved since myfirst visit there about 15 years ago. Pret-a-mangers on every street corner are inexpensive, healthy but a little boring.
We got a kick out of crossing the street in front of the Abbey Rd studios. If you like browsing through knick-knacks, go to Camden market. There was a fire there recently, but I don't know if there was much damage.
There are a number of markets around town. If you're staying in an apartment, they are a good source of food to bring back when you want to eat in.
Take good walking shoes, good rain gear and a good folding umbrella. You're there for the week and don't let the weather spoil your fun.
And if you get asked about politics - or anything else, I hope you engage. I find that to be the best part of traveling.
Enjoy.