Yes cycling could be safer, yes local authorities could do a hell of a lot more to improve things, but that does not mean that cycling is a no no. You might argue anything could be made safer with effort and expense!
I have the same disagreements about cycling safety with my GF, mother and father, my GF's mother gets in on the act too. I don't cycle in London, but I assume it cannot be worse than Bradford where the quality of driving is often lacking and there is very little infrastructure in place for cycling.
Family/spouses use emotional arguments that cannot be reasoned with, no matter the quality of your logic or come backs. They truly don't understand the concept of risk and assume cycling is an instant death warrant as they heard/knew of a guy who was killed or injured on a bike once (hmm). They give little consideration to the dangers of driving, or the thousands who die in their cars every year.
Just do what you want, I know it's hard, the emotional manipulation some people pull to stop you from cycling can be very discouraging (I thought about jacking it in a few times because of the pressure I was under, it gave my confidence a real knock) but their concern is ultimately misplaced and misguided. In the end just get out there and show her that you can do it every day without a hiccup. Just don't go feeding her insecurities with every little mishap, close pass, and fall as it only serves to feed their anxiety about the levels of safety.
Life's too short to be sat in a bus/tube stinking of pee.