I'm a bit unsure exactly what it is you are describing. Pretty much all the crossings I use regularly in London follows an at least very similar pattern to what I'm used to seeing in the US when visiting, with traffic parallel to the crossing getting green at the same time as the pedestrian crossing in question
In fact, here's a discussion in the UK Parliament in 2004 where it was proposed to restrict the use of all-red phases in low traffic junctions, and where the person proposing the change saw it as necessary to describe why this was even an issue:
http://www.publications.parliament.uk/pa/cm200304/cmstand/a/...
"Mr. Redwood: Until a few years ago, no one would have understood why new clause 5 would be necessary. All traffic lights that motorists or cyclists encountered on our highways had a simple form of operation which meant that there was always one free route across the junction showing green. In the normal four-way junction where two roads intersected, one road had priority for one phase and the other road had the alternate priority. There could be more complicated variations at larger junctions, but there was always a green phase for vehicles on one part of the junction"
[The proposed clause 5: 'No authority shall be entitled to install or operate traffic signals with all red phases at road junctions where there is no hazard other than traffic.']
You may be right that no vehicles will have green light to cross the pedestrian crossing while pedestrians have a green light, though I'm not 100% sure that's true either - I seem to remember crossings in London at least where turning traffic will have green at the same time as pedestrians, but I don't drive so maybe that's just me as an impatient pedestrian misremembering, as I'll regularly cross on red.