As of 2 weeks ago Tube stations and national rail accept non chip cards. I bought both one way tickets at the self service and topped up my Oyster card. I had no problems anywhere in London last week of May this year.

I would carry some cash on you in case there are problems. I know people like to insist they are right and get their card run, but if I have the cash I prefer to avoid the hassle, its simply not worth it. With Schwab you can have an ATM card with no forex fees, and a credit card with no forex fees.

Thats just my 2 cents.