Sapman, I am guessing that Uncle dude does not live in London......

London city airport is in the East of the city. London Heathrow is way out to the west of the city. A taxi / mini cab could take up to two hours on a weekday, less at the week-end.... and will easily cost $100.

If you are not lugging lots of heavy luggage, then the docklands railway(connected to city airport) and underground is a good bet, and will cost around $10 (£7.20) You could do this in around 1.5 hours.... It is very easy to use. Have a look at "transport for london" website.

Your thoughts on arriving in London the day before travel back to Phoenix is sensible, and would enable you to drop in to London for a night out (easy to do!)......