This is the best I can do with qplot. Not quite what you requested, but closer. OOOPs, I see, you already understood this.
q <- qplot(x = count, y = reorder(country, count), data = London.melt, geom = "point", facets = medal.type ~.)
Here is a dput version so others can improve:
dput(London.melt) structure(list(country = structure(c(9L, 6L, 3L, 1L, 7L, 4L, 5L, 8L, 2L, 10L, 9L, 6L, 3L, 1L, 7L, 4L, 5L, 8L, 2L, 10L, 9L, 6L, 3L, 1L, 7L, 4L, 5L, 8L, 2L, 10L), .Label = c("Australia", "China", "France", "Germany", "Great Britain & N. Ireland", "Italy", "Japan", "Russian Federation", "South Korea", "United States" ), class = "factor"), medal.type = structure(c(2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L), .Label = c("bronze", "gold", "silver"), class = "factor"), count = c(13L, 8L, 11L, 7L, 7L, 11L, 29L, 24L, 38L, 46L, 8L, 9L, 11L, 16L, 14L, 19L, 17L, 26L, 27L, 29L, 7L, 11L, 12L, 12L, 17L, 14L, 19L, 32L, 23L, 29L)), .Names = c("country", "medal.type", "count"), class = "data.frame", row.names = c("1", "2", "3", "4", "5", "6", "7", "8", "9", "10", "11", "12", "13", "14", "15", "16", "17", "18", "19", "20", "21", "22", "23", "24", "25", "26", "27", "28", "29", "30"))