That's exactly what everyone else wants! How much money you can spend will determine who gets it.
Don't worry too much, your budget seems pretty healthy to me.
Sorry I don't know much about commutes other than Richmond.
Don't get overwhelmed, you can only do so much house hunting from over there. Everything will be a compromise on something so don't get discouraged.
Did I mention that it's really great to live in London? I have driven to France and back in a day, Ireland, Germany and the Czech Republic.