I'm British, and I spent three weeks backpacking around southern Utah / northern Arizona (Paria Canyon through hike, other bits & pieces) in '98. It was absolutely fantastic, trip of a lifetime. Don't just think about the national parks - there's a lot more seriously remote real wilderness out there, e.g. parts of Grand Staircase /Escalante.
I was in the area twice - long trip in June/July, *hot*, regretted bothering to carry a sleeping bag, and short trip at Easter, *snow*, regretted not bringing a down jacket.
The desert is to be taken seriously, especially if you're going into remote areas. If you don't have desert experience, it's worth hooking up with somebody who does for a few days - I spent the first week of my three week visit with a Sierra Club group, some great folks and learned a lot. Might be worth considering something similar.
One specific recommendation for a base area would be around Escalante/Boulder. Idyllic area, heart of Grand Staircase/Escalante National Monument and within easy striking distance of many national parks: Capitol Reef, Bryce, Zion. Close to mountains if the desert's too hot, close to desert if the mountains are too cold.