I'm a little out of date, not having lived in London for a little while now, but unless anything new and great has come out:

London Travel for pretty much all your London travel needs. Not a pretty UI, though. Free, no hideous permissions as far as I remember.

I also use something like London Bus Checker. I got the full-UK version without ads as the Amazon Free App Of The Day a while back. That link is for the London-Only one, and I think it's ad-supported. Not having used that I don't know how obtrusive the ads are.
It's... quite neat, but I don't know that I'd pay for it.

HTH :-)