An old American friend, Lord Affectation (I've got other less flattering names for him too), has moved to Kent and informed me that "The British call bicycles push irons!" I said, "Your English friends are pushing your leg." Most English people call them bicycles, just like the rest of the English-speaking world. But as the photo (taken near a bike rack on campus) shows, he's partly right. Some people do call them push bikes, including The Gardener (Better Half's Sister's Better Half--BH just called him a young fogey). It's a little old-fashioned, but it succeeds in distinguishing them from motorbikes. AmE sometimes uses pedal bike for this purpose, but bicycle would be more usual. Or just a sign saying No M/C.
The OED has push bicycle and push cycle as well as push bike, which they label as 'informal.' Myself, I've only heard push bike, but push bicycle is around on the web, as well as push tricycle and push unicycle. The OED does not have push iron, but the web holds a small number of examples, some of them in lists of Wigan dialect words. Wigan, it should be noted, is nowhere near Kent.
Of course, BrE has another push-vehicle, the push-chair, known in AmE as a stroller. Related is the pram, short for perambulator, known in AmE as a baby carriage.