That would probably work for you. I assume, originating in the U.K. that you would fly into New York or some other east coast destination and then on to Las Vegas. (looks like Virgin Atlantic is the only carrier offering non-stop LON-LAS right now). At any rate, on arriving in LAS keep in mind that the hotels on the "strip" are very expensive and you don't actually spend much time in the hotel room. I've had good luck with one of three hotels in LAS all run by Sun Coast and just off the strip (see here:
http://maps.google.com/maps?hl=en&resnum=0&um= 1&ie=UTF-8&q=SunCoast+hotels&near=Las+Vegas,+N V&fb=1&view=text&sa=X&oi=local_group&resnum=4& ct=more-results&cd=1 (apologies for the length URL) .
Keep in mind the drive from Vegas to Grand Canyon is about 5 or 6 hours to the South Rim. Highways are generally very good, with the exception of the Hoover Dam area which can be quite congested. The highway to Kingman, AZ is two lane, but still a very good road. Keep in mind the altitude on the south rim is above 5,000 feet above sea level and can be wuite cool, even in some summer days. I would plan on staying in Flagstaff, rather than hotel near Grand Canyon. A good chain here in the U.S. for value is Comfort Inns. Your exchange rate is certainly favorable right now.
The drive from Gand Canyon to LA will be lengthy. To avoid repeating the LAS drive, you can go south from South Rim via Flagstaff and Sedona which is very scenic to Phoenix, thence west through Blythe, CA and returning to LA. If you thought LA to Las Vegas was boring, the Mojave Desert in the area of Blythe/ Twenty Nine Palms exceeds that by a whole bunch.
I'd look at Map Quest as well as Google some information on where to stay. Personally, I'd plan on staying in Phoeni